text: 17 (cid:48) ≤ | (cid:48) − | ≤ Similarly, we can show that C(MP ) C(MP) + (cid:15). This shows that C(MP ) C(MP) (cid:15) for every (cid:48) ∈ T MP UMP. Therefo≤re, C(cid:40)≤is continu(cid:41)ous. But C(Wˆ ) = C(MPW×ˆ )M, soPCXis c→ontRinuous on (DMC(Xo,)∗, X(o,∗)). Now for every 0 i (cid:89)n, deﬁne the mapping f : ∆iX ( ) backward-recursively as i follows: n • f (pn, MP) = max p (x ) . n 1 ∈C i i (cid:90) xn ≤ 1 i=1 • For every 0 i < n, deﬁne · f (pi , MP) = f (pi+1, MP) dMP(p ). i 1 n 1 i+1 ∆X ≤ Clearly f is continuous. Now let 0 i <(cid:90) n and assume that f is continuous. If we let S = × MP nX × MP Xi+1× MP X ∆iX ( ), Lemma 5 implies that the mapping F : ∆iX ( ) ( ) deﬁned as i (cid:48) · (cid:48) F (pi , MP, MP ) = f(pi+1, MP) dMP (p ) i 1 1 i+1 ∆X is continuous. But f (pi , MP) = F (pi , MP, MP), so f is also continuous. Therefore, f is continuous. i 1 |Xi| 1 i 0 − n T By noticing that P C(Wˆ ) = 1 |C| f (MP ), we conclude that P C is continuous on (DMC(Xo)∗, X(o∗)). e, 0 Wˆ e, , , Moreover, since P is the minimum of a ﬁnite family of continuous mappings, it is continuous. e,n,M It is worth mentioning that Proposition 6 can be shown from Proposition 9 because the noisiness topology is coarser than the strong topology. Corollary 4. All the mappings in Proposition 9 are also continuous if we replace the noisiness topology T T X(o∗) with the total variation topology (o)X ∗. , TV, , T T Proof. This is true because (o)X ∗ is ﬁner than X(o∗). TV, , , B. Channel operations In the following, we show that we can express the channel operations in terms of Blackwell measures. We have all the tools to achieve this for the channel sum, channel product and channel interpolation. In order to express the channel polarization transformations in terms of the Blackwell measures, we need to introduce new deﬁnitions. X ∗ X ∗ Let be a ﬁnite set and let be a binary operation on a ﬁnite set . We say that is uniformity → ∗ X ∈ X preserving if the mapping (a, b) (a b, b) is a bijection from 2 to itself [16]. For every a, b , we ∈ X ∗ ∗ ∗ denote the unique element c satisfying c b = a as c = a/ b. Note that / is a binary operation and ∗ ∗ it is uniformity preserving. / is called the right-inverse of . It was shown in [9] that a binary operation is polarizing if and only if it is uniformity preserving and its inverse is strongly ergodic. Binary operations that are not uniformity preserving are not interesting for polarization theory because they do not preserve the symmetric capacity [9]. Therefore, we will only focus on polarization transformations that are based on uniformity preserv(cid:88)ing operations. ∗ X − ∗ × → Let be a ﬁxed uniformity preserving operation on . Deﬁne the mapping C , : ∆X ∆X ∆X as − ∗ ∗ (C , (p , p ))(u ) = p (u u )p (u ). 1 2 1 1 1 2 2 2 ∈X u2 − ∗ The probability distribution C , (p , p ) can be interpreted as follows: let X and X be two independent X 1 2 1 2 random variables in that are distributed as p and p respectively, and let (U , U ) be the random pair X ∗ 1 2 ∗ 1 2 − ∗ in 2 deﬁned as (U , U ) = (X / X , X ), or equivalently (X , X ) = (U U , U ). C , (p , p ) is the 1 2 1 2 2 1 2 1 2 2 1 2 probability distribution of U . 1 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Similarly,

text: we

text: can

text: show

text: that

text: C(MP

text: C(MP)

text: This

text: shows

text: that

text: C(MP

text: C(MP)

text: for

text: every

text: T

text: MP

text: UMP.

text: Therefo≤re,

label: list

text: But

text: C(Wˆ

text: C(MPW×ˆ

text: )M,

text: soPCXis

text: c→ontRinuous

text: on

text: (DMC(Xo,)∗,

text: X(o,∗)).

text: Now

text: for

text: every

text: i

text: deﬁne

text: the

text: mapping

text: f

text: ∆iX

text: backward-recursively

text: as

text: i

text: follows:

text: n

text: f

text: (pn,

text: MP)

text: max

text: p

text: (x

text: n

text: ∈C

text: i

text: i

text: xn

text: For

text: every

text: i

text: n,

text: deﬁne

text: f

text: (pi

text: MP)

text: f

text: MP)

text: dMP(p

text: i

text: n

text: ∆X

text: Clearly

text: f

text: is

text: continuous.

text: Now

text: let

text: i

text: n

text: and

text: assume

text: that

text: f

text: is

text: continuous.

text: If

text: we

text: let

text: S

text: MP

text: nX

text: MP

text: MP

text: X

text: ∆iX

text: Lemma

text: implies

text: that

text: the

text: mapping

text: F

text: ∆iX

text: deﬁned

text: as

text: i

text: F

text: (pi

text: MP,

text: MP

text: MP)

text: dMP

text: (p

text: i

text: ∆X

text: is

text: continuous.

text: But

text: f

text: (pi

text: MP)

text: F

text: (pi

text: MP,

text: MP),

text: so

text: f

text: is

text: also

text: continuous.

text: Therefore,

text: f

text: is

text: continuous.

text: i

text: |Xi|

text: i

text: n

text: T

text: By

text: noticing

text: that

text: P

text: C(Wˆ

text: |C|

text: f

text: (MP

text: we

text: conclude

text: that

text: P

text: C

text: is

text: continuous

text: on

text: (DMC(Xo)∗,

text: X(o∗)).

text: e,

text: Wˆ

text: e,

text: Moreover,

text: since

text: P

text: is

text: the

text: minimum

text: of

text: a

text: ﬁnite

text: family

text: of

text: continuous

text: mappings,

text: it

text: is

text: continuous.

text: e,n,M

text: It

text: is

text: worth

text: mentioning

text: that

text: Proposition

text: can

text: be

text: shown

text: from

text: Proposition

text: because

text: the

text: noisiness

text: topology

text: is

text: coarser

text: than

text: the

text: strong

text: topology.

text: Corollary

text: All

text: the

text: mappings

text: in

text: Proposition

text: are

text: also

text: continuous

text: if

text: we

text: replace

text: the

text: noisiness

text: topology

text: T

text: T

text: X(o∗)

text: with

text: the

text: total

text: variation

text: topology

text: (o)X

text: TV,

text: T

text: T

text: Proof.

text: This

text: is

text: true

text: because

text: (o)X

text: is

text: ﬁner

text: than

text: X(o∗).

text: TV,

text: B.

text: Channel

text: operations

text: In

text: the

text: following,

text: we

text: show

text: that

text: we

text: can

text: express

text: the

text: channel

text: operations

text: in

text: terms

text: of

text: Blackwell

text: measures.

text: We

text: have

text: all

text: the

text: tools

text: to

text: achieve

text: this

text: for

text: the

text: channel

text: sum,

text: channel

text: product

text: and

text: channel

text: interpolation.

text: In

text: order

text: to

text: express

text: the

text: channel

text: polarization

text: transformations

text: in

text: terms

text: of

text: the

text: Blackwell

text: measures,

text: we

text: need

text: to

text: introduce

text: new

text: deﬁnitions.

text: X

text: X

text: Let

text: be

text: a

text: ﬁnite

text: set

text: and

text: let

text: be

text: a

text: binary

text: operation

text: on

text: a

text: ﬁnite

text: set

text: We

text: say

text: that

text: is

text: uniformity

text: X

text: X

text: preserving

text: if

text: the

text: mapping

text: (a,

text: b)

text: (a

text: b,

text: b)

text: is

text: a

text: bijection

text: from

text: to

text: itself

text: For

text: every

text: a,

text: b

text: we

text: X

text: denote

text: the

text: unique

text: element

text: c

text: satisfying

text: c

text: b

text: a

text: as

text: c

text: a/

text: b.

text: Note

text: that

text: is

text: a

text: binary

text: operation

text: and

text: it

text: is

text: uniformity

text: preserving.

text: is

text: called

text: the

text: right-inverse

text: of

text: It

text: was

text: shown

text: in

text: that

text: a

text: binary

text: operation

text: is

text: polarizing

text: if

text: and

text: only

text: if

text: it

text: is

text: uniformity

text: preserving

text: and

text: its

text: inverse

text: is

text: strongly

text: ergodic.

text: Binary

text: operations

text: that

text: are

text: not

text: uniformity

text: preserving

text: are

text: not

text: interesting

text: for

text: polarization

text: theory

text: because

text: they

text: do

text: not

text: preserve

text: the

text: symmetric

text: capacity

text: Therefore,

text: we

text: will

text: only

text: focus

text: on

text: polarization

text: transformations

text: that

text: are

text: based

text: on

text: uniformity

text: operations.

text: X

text: Let

text: be

text: a

text: ﬁxed

text: uniformity

text: preserving

text: operation

text: on

text: Deﬁne

text: the

text: mapping

text: C

text: ∆X

text: ∆X

text: ∆X

text: as

text: (C

text: (p

text: p

text: ))(u

text: p

text: (u

text: u

text: )p

text: (u

text: ∈X

text: The

text: probability

text: distribution

text: C

text: (p

text: p

text: can

text: be

text: interpreted

text: as

text: follows:

text: let

text: X

text: and

text: X

text: be

text: two

text: independent

text: X

text: random

text: variables

text: in

text: that

text: are

text: distributed

text: as

text: p

text: and

text: p

text: respectively,

text: and

text: let

text: (U

text: U

text: be

text: the

text: random

text: pair

text: X

text: in

text: deﬁned

text: as

text: (U

text: U

text: (X

text: X

text: X

text: or

text: equivalently

text: (X

text: X

text: (U

text: U

text: U

text: C

text: (p

text: p

text: is

text: the

text: probability

text: distribution

text: of

text: U

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 13 In [3], equivalent representations for binary memoryless symmetric (BMS) channels (namely L, D and G densities) were provided. A necessary and sufﬁcient condition for the degradation of a BMS channel (cid:48) | | (cid:48) W with respect to another BMS channel W was given in [3] in terms of the D -densities of W and W . It immediately follows from this condition that two BMS channels are equivalent if and only if they have | | the same D -densities. One can deduce from this that two BMS channels (with ﬁnite output alphabets) are equivalent if and only if they have the same Blackwell measure. The following proposition shows that this is also true for channels with arbitrary (but ﬁnite) input and output alphabets: X Y Z ∈ (cid:48) ∈ Proposition 2. [15] Let , and be three ﬁnite sets. Two channels W DMCX Y and W DMCX Z , , are equivalent if and only if MP = MP (cid:48). W W Proof. This proposition is known [15], but we provide a proof in Appendix for completeness. ∈ |Z| Corollary 1. If W DMCX Y and rank(W ) > , then W is not equivalent to any channel in DMCX Z. , , (cid:48) | | ≤ |Z| (cid:48) ∈ Proof. Since rank(W ) = supp(MP (cid:48)) for every W DMCX Z, it is impossible for W to be (cid:48) W , equivalent to any channel W in DMCX Z. , |X | X Corollary 2. If = 1, all channels with input alphabet are equivalent. X Y VI. SPACE OF EQUIVALENT CHANNELS FROM TO A. The DMC(Xo)Y space , X Y Let and be two ﬁnite sets. Deﬁne the relation RX(o)Y on DMCX Y as follows: , , ∀ (cid:48) ∈ (cid:48) ⇔ (cid:48) W, W DMCX Y, W RX(o)YW W is equivalent to W . , , It is easy to see that RX(o)Y is an equivalence relation on DMCX Y. , , X Y Deﬁnition 1. The space of equivalent channels with input alphabet and output alphabet is the X Y quotient of the space of channels from to by the equivalence relation: DMC(Xo)Y = DMCX Y /RX(o)Y. , , , T T We deﬁne the topology X(oY) on DMC(Xo)Y as the quotient topology X Y/RX(o)Y. , , , , T Unless we explicitly state otherwise, we always associate DMC(Xo)Y with the quotient topology X(oY) . ∈ ∈ , , For every W DMCX Y, let Wˆ DMC(Xo)Y be the RX(o)Y-equivalence class containing W . , , , → Lemma 3. The projection mapping Proj : DMCX Y DMC(Xo)Y deﬁned as Proj(W ) = Wˆ is continuous , , and closed. Proof. See Appendix . ∈ Corollary 3. For every W DMCX Y, Wˆ is a compact subset of DMCX Y. , , Proof. Since DMCX Y is compact, then DMC(Xo)Y = DMCX Y /RX(o)Y is compact as well. , → , , , { } Let Proj : DMCX Y DMC(Xo)Y be as in Lemma 3. Since Proj is closed and since W is closed in { } , { } , − { } DMCX Y, Wˆ = Proj( W ) is closed in DMC(Xo)Y. Therefore, Wˆ = Proj 1( Wˆ ) is closed in DMCX Y , , , because Proj is continuous. Now since DMCX Y is compact, Wˆ is compact as well. , Theorem 3. DMC(Xo)Y is a compact, path-connected and metrizable space. , Proof. Since DMCX Y is compact and path-connected, DMC(Xo)Y = DMCX Y /RX(o)Y is compact and path- , , , , connected as well. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: In

text: equivalent

text: representations

text: for

text: binary

text: memoryless

text: symmetric

text: (BMS)

text: channels

text: (namely

text: L,

text: D

text: and

text: G

text: densities)

text: were

text: provided.

text: A

text: necessary

text: and

text: sufﬁcient

text: condition

text: for

text: the

text: degradation

text: of

text: a

text: BMS

text: channel

text: W

text: with

text: respect

text: to

text: another

text: BMS

text: channel

text: W

text: was

text: given

text: in

text: in

text: terms

text: of

text: the

text: D

text: -densities

text: of

text: W

text: and

text: W

text: It

text: immediately

text: follows

text: from

text: this

text: condition

text: that

text: two

text: BMS

text: channels

text: are

text: equivalent

text: if

text: and

text: only

text: if

text: they

text: have

text: the

text: same

text: D

text: -densities.

text: One

text: can

text: deduce

text: from

text: this

text: that

text: two

text: BMS

text: channels

text: (with

text: ﬁnite

text: output

text: alphabets)

text: are

text: equivalent

text: if

text: and

text: only

text: if

text: they

text: have

text: the

text: same

text: Blackwell

text: measure.

text: The

text: following

text: proposition

text: shows

text: that

text: this

text: is

text: also

text: true

text: for

text: channels

text: with

text: arbitrary

text: (but

text: ﬁnite)

text: input

text: and

text: output

text: alphabets:

text: X

text: Y

text: Z

text: Proposition

text: Let

text: and

text: be

text: three

text: ﬁnite

text: sets.

text: Two

text: channels

text: W

text: DMCX

text: Y

text: and

text: W

text: DMCX

text: Z

text: are

text: equivalent

text: if

text: and

text: only

text: if

text: MP

text: MP

text: W

text: W

text: Proof.

text: This

text: proposition

text: is

text: known

text: but

text: we

text: provide

text: a

text: proof

text: in

text: Appendix

text: for

text: completeness.

text: |Z|

text: Corollary

text: If

text: W

text: DMCX

text: Y

text: and

text: rank(W

text: then

text: W

text: is

text: not

text: equivalent

text: to

text: any

text: channel

text: in

text: DMCX

text: Z.

text: |Z|

text: Proof.

text: Since

text: rank(W

text: supp(MP

text: for

text: every

text: W

text: DMCX

text: Z,

text: it

text: is

text: impossible

text: for

text: W

text: to

text: be

text: W

text: equivalent

text: to

text: any

text: channel

text: W

text: in

text: DMCX

text: Z.

text: |X

text: X

text: Corollary

text: If

text: all

text: channels

text: with

text: input

text: alphabet

text: are

text: equivalent.

text: X

text: Y

text: VI.

text: SPACE

text: OF

text: EQUIVALENT

text: CHANNELS

text: FROM

text: TO

text: A.

text: The

text: DMC(Xo)Y

text: space

text: X

text: Y

text: Let

text: and

text: be

text: two

text: ﬁnite

text: sets.

text: Deﬁne

text: the

text: relation

text: RX(o)Y

text: on

text: DMCX

text: Y

text: as

text: follows:

text: W,

text: W

text: DMCX

text: Y,

text: W

text: RX(o)YW

text: W

text: is

text: equivalent

text: to

text: W

text: It

text: is

text: easy

text: to

text: see

text: that

text: RX(o)Y

text: is

text: an

text: equivalence

text: relation

text: on

text: DMCX

text: Y.

text: X

text: Y

text: Deﬁnition

text: The

text: space

text: of

text: equivalent

text: channels

text: with

text: input

text: alphabet

text: and

text: output

text: alphabet

text: is

text: the

text: X

text: Y

text: quotient

text: of

text: the

text: space

text: of

text: channels

text: from

text: to

text: by

text: the

text: equivalence

text: relation:

text: DMC(Xo)Y

text: DMCX

text: Y

text: /RX(o)Y.

text: T

text: T

text: We

text: deﬁne

text: the

text: topology

text: X(oY)

text: on

text: DMC(Xo)Y

text: as

text: the

text: quotient

text: topology

text: X

text: Y/RX(o)Y.

text: T

text: Unless

text: we

text: explicitly

text: state

text: otherwise,

text: we

text: always

text: associate

text: DMC(Xo)Y

text: with

text: the

text: quotient

text: topology

text: X(oY)

text: For

text: every

text: W

text: DMCX

text: Y,

text: let

text: Wˆ

text: DMC(Xo)Y

text: be

text: the

text: RX(o)Y-equivalence

text: class

text: containing

text: W

text: Lemma

text: The

text: projection

text: mapping

text: Proj

text: DMCX

text: Y

text: DMC(Xo)Y

text: deﬁned

text: as

text: Proj(W

text: Wˆ

text: is

text: continuous

text: and

text: closed.

text: Proof.

text: See

text: Appendix

text: Corollary

text: For

text: every

text: W

text: DMCX

text: Y,

text: Wˆ

text: is

text: a

text: compact

text: subset

text: of

text: DMCX

text: Y.

text: Proof.

text: Since

text: DMCX

text: Y

text: is

text: compact,

text: then

text: DMC(Xo)Y

text: DMCX

text: Y

text: /RX(o)Y

text: is

text: compact

text: as

text: well.

text: Let

text: Proj

text: DMCX

text: Y

text: DMC(Xo)Y

text: be

text: as

text: in

text: Lemma

text: Since

text: Proj

text: is

text: closed

text: and

text: since

text: W

text: is

text: closed

text: in

text: DMCX

text: Y,

text: Wˆ

text: Proj(

text: W

text: is

text: closed

text: in

text: DMC(Xo)Y.

text: Therefore,

text: Wˆ

text: Proj

text: Wˆ

text: is

text: closed

text: in

text: DMCX

text: Y

text: because

text: Proj

text: is

text: continuous.

text: Now

text: since

text: DMCX

text: Y

text: is

text: compact,

text: Wˆ

text: is

text: compact

text: as

text: well.

text: Theorem

text: DMC(Xo)Y

text: is

text: a

text: compact,

text: path-connected

text: and

text: metrizable

text: space.

text: Proof.

text: Since

text: DMCX

text: Y

text: is

text: compact

text: and

text: path-connected,

text: DMC(Xo)Y

text: DMCX

text: Y

text: /RX(o)Y

text: is

text: compact

text: and

text: path-

text: connected

text: as

text: well.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 18 (cid:32) (cid:33) (cid:92) (cid:16) (cid:17) (cid:91) Now since \ \ DMC(Xo)∗ DMC(Xo) = DMC(Xo)∗ DMC(Xo) = ø, , ,[n] , ,[n] ≥ ≥ n 1 n 1 \ T ≥ and since DMC(Xo)∗ DMC(Xo) is open and dense in (DMC(Xo)∗, ) for every n 1, we conclude that T , ,[n] , (DMC(Xo)∗, ) is not a Baire space. , |X | ≥ Corollary 7. If 2, no natural topology on DMC(Xo)∗ can be completely metrizable. , Proof. The corollary follows from Proposition 7 and the fact that every completely metrizable topology is both Hausdorff and Baire. |X | ≥ T T Proposition 8. If 2 and is a Hausdorff natural topology, then (DMC(Xo)∗, ) is not locally compact ∈ , T anywhere, i.e., for every Wˆ DMC(Xo)∗, there is no compact neighborhood of Wˆ in (DMC(Xo)∗, ). , , Proof. Assume to the contrary that there exists a compact neighborhood K of Wˆ . There exists an open ∈ ⊂ set U such that Wˆ U K. Since K is compact and Hausdorff, it is a Baire space. Moreover, since U is an open subset of K, U is also a Baire space. ≥ T ∩ Fix n 1. Since the interior of DMC(Xo) in (DMC(Xo)∗, ) is empty, the interior of U DMC(Xo) in \ ,[n] , T ,[n] U is also empty. Therefore, U DMC(Xo) is dense in U. On the other hand, since is natural, DMC(Xo) is compact which implies th(cid:92)at (cid:16)it is clo,s[ne]d bec(cid:17)ause T i(cid:32)s H(cid:91)ausdorff. Th(cid:33)erefore, U \ DMC(Xo) is open,[inn] ,[n] U. Now since \ \ U DMC(Xo) = U DMC(Xo) = ø, ,[n] ,[n] ≥ ≥ n 1 n 1 \ ≥ and since U DMC(Xo) is open and dense in U for every n 1, U is not Baire, which is a contradiction. ,[n] T Therefore, there is no compact neighborhood of Wˆ in (DMC(Xo)∗, ). , VIII. STRONG TOPOLOGY ON DMC(Xo)∗ , T The ﬁrst natural topology that we study is the strong topology (Xo) ∗ on DMC(Xo)∗, which is the ﬁnest s, , , natural topology. { } (cid:77) Since the spaces DMCX ≥ are disjoint and since there is no a priori way to (topologically) compare ,[n] n 1 (cid:54) (cid:48) channels in DMCX with channels in DMCX (cid:48) for n = n , the “most natural” topology that we can ,[n] ,[nT] T T deﬁne on DMCX ∗ is the disjoint union topology X ∗ := X . Clearly, the space (DMCX ∗, X ∗) , s, , ,[n] , s, , ≥ T n 1 is disconnected. Moreover, X ∗ is metrizable because it is the disjoint union of metrizable spaces. It is s, , also σ-compact because it is the union of countably many compact spaces. T We added the subscript s to emphasize the fact that X ∗ is a strong topology (remember that the s, , disjoint union topology is the ﬁnest topology that makes the canonical injections continuous). T T Deﬁnition 4. We deﬁne the strong topology (Xo) ∗ on DMC(Xo)∗ as the quotient topology X ∗/RX(o)∗. Ts, , , s, , , We call open and closed sets in (DMC(Xo)∗, (Xo) ∗) as strongly open and strongly closed sets respectively. , s, , → Let Proj : DMCX ∗ DMC(Xo)∗ be the projection onto the RX(o)∗-equivalence classes, and for every ≥ , → , , n 1 let Proj : DMCX DMC(Xo) be the projection onto the RX(o) -equivalence classes. Due to n ,[n] ,[n] ,[n] ∈ the identiﬁcations that we made in Section , (cid:97)we have Proj(W ) = Proj (W ) for every W DMCX . ⊂ n ,[n] Therefore, for every U DMC(Xo)∗, we have , − − ∩ Proj 1(U) = Proj 1(U DMC(Xo) ). n ,[n] ≥ n 1 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Now

text: since

text: DMC(Xo)∗

text: DMC(Xo)

text: DMC(Xo)∗

text: DMC(Xo)

text: ø,

text: ,[n]

text: ,[n]

text: n

text: n

text: T

text: and

text: since

text: DMC(Xo)∗

text: DMC(Xo)

text: is

text: open

text: and

text: dense

text: in

text: (DMC(Xo)∗,

text: for

text: every

text: n

text: we

text: conclude

text: that

text: T

text: ,[n]

text: (DMC(Xo)∗,

text: is

text: not

text: a

text: Baire

text: space.

text: |X

text: Corollary

text: If

text: no

text: natural

text: topology

text: on

text: DMC(Xo)∗

text: can

text: be

text: completely

text: metrizable.

text: Proof.

text: The

text: corollary

text: follows

text: from

text: Proposition

text: and

text: the

text: fact

text: that

text: every

text: completely

text: metrizable

text: topology

text: is

text: both

text: Hausdorff

text: and

text: Baire.

text: |X

text: T

text: T

text: Proposition

text: If

text: and

text: is

text: a

text: Hausdorff

text: natural

text: topology,

text: then

text: (DMC(Xo)∗,

text: is

text: not

text: locally

text: compact

text: T

text: anywhere,

text: i.e.,

text: for

text: every

text: Wˆ

text: DMC(Xo)∗,

text: there

text: is

text: no

text: compact

text: neighborhood

text: of

text: Wˆ

text: in

text: (DMC(Xo)∗,

text: Proof.

text: Assume

text: to

text: the

text: contrary

text: that

text: there

text: exists

text: a

text: compact

text: neighborhood

text: K

text: of

text: Wˆ

text: There

text: exists

text: an

text: open

text: set

text: U

text: such

text: that

text: Wˆ

text: U

text: K.

text: Since

text: K

text: is

text: compact

text: and

text: Hausdorff,

text: it

text: is

text: a

text: Baire

text: space.

text: Moreover,

text: since

text: U

text: is

text: an

text: open

text: subset

text: of

text: K,

text: U

text: is

text: also

text: a

text: Baire

text: space.

text: T

text: Fix

text: n

text: Since

text: the

text: interior

text: of

text: DMC(Xo)

text: in

text: (DMC(Xo)∗,

text: is

text: empty,

text: the

text: interior

text: of

text: U

text: DMC(Xo)

text: in

text: ,[n]

text: T

text: ,[n]

text: U

text: is

text: also

text: empty.

text: Therefore,

text: U

text: DMC(Xo)

text: is

text: dense

text: in

text: U.

text: On

text: the

text: other

text: hand,

text: since

text: is

text: natural,

text: DMC(Xo)

text: is

text: compact

text: which

text: implies

text: is

text: clo,s[ne]d

text: T

text: U

text: DMC(Xo)

text: is

text: open,[inn]

text: ,[n]

text: U.

text: Now

text: since

text: U

text: DMC(Xo)

text: U

text: DMC(Xo)

text: ø,

text: ,[n]

text: ,[n]

text: n

text: n

text: and

text: since

text: U

text: DMC(Xo)

text: is

text: open

text: and

text: dense

text: in

text: U

text: for

text: every

text: n

text: U

text: is

text: not

text: Baire,

text: which

text: is

text: a

text: contradiction.

text: ,[n]

text: T

text: Therefore,

text: there

text: is

text: no

text: compact

text: neighborhood

text: of

text: Wˆ

text: in

text: (DMC(Xo)∗,

text: VIII.

text: STRONG

text: TOPOLOGY

text: ON

text: DMC(Xo)∗

text: T

text: The

text: ﬁrst

text: natural

text: topology

text: that

text: we

text: study

text: is

text: the

text: strong

text: topology

text: (Xo)

text: on

text: DMC(Xo)∗,

text: which

text: is

text: the

text: ﬁnest

text: s,

text: natural

text: topology.

text: Since

text: the

text: spaces

text: DMCX

text: are

text: disjoint

text: and

text: since

text: there

text: is

text: no

text: a

text: priori

text: way

text: to

text: (topologically)

text: compare

text: ,[n]

text: n

text: channels

text: in

text: DMCX

text: with

text: channels

text: in

text: DMCX

text: for

text: n

text: n

text: the

text: “most

text: natural”

text: topology

text: that

text: we

text: can

text: ,[n]

text: ,[nT]

text: T

text: T

text: deﬁne

text: on

text: DMCX

text: is

text: the

text: disjoint

text: union

text: topology

text: X

text: X

text: Clearly,

text: the

text: space

text: (DMCX

text: X

text: s,

text: ,[n]

text: s,

text: T

text: n

text: is

text: disconnected.

text: Moreover,

text: X

text: is

text: metrizable

text: because

text: it

text: is

text: the

text: disjoint

text: union

text: of

text: metrizable

text: spaces.

text: It

text: is

text: s,

text: also

text: σ-compact

text: because

text: it

text: is

text: the

text: union

text: of

text: countably

text: many

text: compact

text: spaces.

text: T

text: We

text: added

text: the

text: subscript

text: s

text: to

text: emphasize

text: the

text: fact

text: that

text: X

text: is

text: a

text: strong

text: topology

text: (remember

text: that

text: the

text: s,

text: disjoint

text: union

text: topology

text: is

text: the

text: ﬁnest

text: topology

text: that

text: makes

text: the

text: canonical

text: injections

text: continuous).

text: T

text: T

text: Deﬁnition

text: We

text: deﬁne

text: the

text: strong

text: topology

text: (Xo)

text: on

text: DMC(Xo)∗

text: as

text: the

text: quotient

text: topology

text: X

text: ∗/RX(o)∗.

text: Ts,

text: s,

text: We

text: call

text: open

text: and

text: closed

text: sets

text: in

text: (DMC(Xo)∗,

text: (Xo)

text: as

text: strongly

text: open

text: and

text: strongly

text: closed

text: sets

text: respectively.

text: s,

text: Let

text: Proj

text: DMCX

text: DMC(Xo)∗

text: be

text: the

text: projection

text: onto

text: the

text: RX(o)∗-equivalence

text: classes,

text: and

text: for

text: every

text: n

text: let

text: Proj

text: DMCX

text: DMC(Xo)

text: be

text: the

text: projection

text: onto

text: the

text: RX(o)

text: -equivalence

text: classes.

text: Due

text: to

text: n

text: ,[n]

text: ,[n]

text: ,[n]

text: the

text: identiﬁcations

text: that

text: we

text: made

text: in

text: Section

text: have

text: Proj(W

text: Proj

text: (W

text: for

text: every

text: W

text: DMCX

text: n

text: ,[n]

text: Therefore,

text: for

text: every

text: U

text: DMC(Xo)∗,

text: we

text: have

text: Proj

text: Proj

text: DMC(Xo)

text: n

text: ,[n]

text: n

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 A. Set-theoretic notations { } For every integer n > 0, we denote the set 1, . . . , n as [n]. The set of mappings from a set A to a set B is de(cid:40)noted as BA. → { } Let A be a subset of B. The indicator mapping 1 : B 0, 1 of A in B is deﬁned as: A,B ∈ 1 if x A, 1 (x) = 1 ∈ = A,B x A 0 otherwise. If the superset B is clear from the context, we simply write 1 to denote the indicator mapping of A in A B. The power set of B is the set of subsets of B. Since every subset of B can be identiﬁed with its { } indicator mapping, we denote the power set of B as 2B := 0, 1 B. A ⊂ A(cid:48) ⊂ A(cid:48) ⊂ A A collection 2B of subsets of B is said to be ﬁner than another collection 2B if . If (cid:97) (cid:91) A(cid:48) A (cid:97) this is the case, we also say that is coarser than . Let (A ) ∈ be a collection of arbitrary sets indexed by I. The disjoint union of (A ) ∈ is deﬁned as i i I × { } ∈ i i I → Ai = (Ai i ). For every i I, the ith-canonical i(cid:97)njection is the mapping φi : Ai Aj ∈ ∈ ∈ i I i I × { } j I deﬁned as φ (x ) = (x , i). If no confusions can arise, we can identify A with A i through the i i i i ∈ i canonical injection. Therefore, we can see A as a subset of A for every i I. i j ∈ × j I∈ ∈ A relation R on a set T is a subset of T T . For every x, y T , we write xRy to denote (x, y) R. ∈ A relation is said to be reﬂexive if xRx for every x T . It is symmetric if xRy implies yRx for every ∈ ∈ x, y T . It is anti-symmetric if xRy and yRx imply x = y for every x, y T . It is transitive if xRy ∈ and yRz imply xRz for every x, y, z T . An order relation is a relation that is reﬂexive, anti-symmetric and transitive. An equivalence relation is a relation that is reﬂexive, symmetric and transitive. ∈ { ∈ } Let R be an equivalence relation on T . For every x T , the set xˆ = y T : xRy is the R-equivalence class of x. The collection of R-equivalence classes, which we denote as T/R, forms a → partition of T , and it is called the quotient space of T by R. The mapping Proj : T T/R deﬁned as ∈ R Proj (x) = xˆ for every x T is the projection mapping onto T/R. R B. Topological spaces U U ⊂ A topological space is a pair (T, ), where 2T is a collection of subsets of T satisfying: ∈ U ∈ U • ø and T . U U • The intersection of a ﬁnite collection of members of is also a member of . U U • The union of an arbitrary collection of members of is also a member of . U U If (T, ) is a topological space, we say that is a topology on T . The power set 2T of T is clearly a topology. It is called the discrete topology on T . A (cid:92) A If is a an arbitrary collection of subsets of T , we can construct a topology on T starting from as follows: V . A⊂V⊂ V 2T, is a topology on T A A This is the coarsest topology on T that contains . It is called the topology on T generated by . U U Let (T, ) be a topological space. The subsets of T that are members of are called the open sets of T . Complements of open sets are called closed sets. We can easily see that the closed sets satisfy the following: • ø and T are closed. • The union of a ﬁnite collection of closed sets is closed. • The intersection of an arbitrary collection of closed sets is closed.

text: A.

label: section

text: Set-theoretic

text: notations

text: For

text: every

text: integer

text: n

text: we

text: denote

text: the

text: set

text: n

text: as

text: [n].

text: The

text: set

text: of

text: mappings

text: from

text: a

text: set

text: A

text: to

text: a

text: set

text: B

text: is

text: as

text: BA.

text: Let

text: A

text: be

text: a

text: subset

text: of

text: B.

text: The

text: indicator

text: mapping

text: B

text: of

text: A

text: in

text: B

text: is

text: deﬁned

text: as:

text: A,B

text: if

text: x

text: A,

text: (x)

text: A,B

text: x

text: A

text: otherwise.

text: If

text: the

text: superset

text: B

text: is

text: clear

text: from

text: the

text: context,

text: we

text: simply

text: write

text: to

text: denote

text: the

text: indicator

text: mapping

text: of

text: A

text: in

text: A

text: B.

text: The

text: power

text: set

text: of

text: B

text: is

text: the

text: set

text: of

text: subsets

text: of

text: B.

text: Since

text: every

text: subset

text: of

text: B

text: can

text: be

text: identiﬁed

text: with

text: its

text: indicator

text: mapping,

text: we

text: denote

text: the

text: power

text: set

text: of

text: B

text: as

text: B.

text: A

text: A

text: A

text: collection

text: of

text: subsets

text: of

text: B

text: is

text: said

text: to

text: be

text: ﬁner

text: than

text: another

text: collection

text: if

text: If

text: A

text: this

text: is

text: the

text: case,

text: we

text: also

text: say

text: that

text: is

text: coarser

text: than

text: Let

text: (A

text: be

text: a

text: collection

text: of

text: arbitrary

text: sets

text: indexed

text: by

text: I.

text: The

text: disjoint

text: union

text: of

text: (A

text: is

text: deﬁned

text: as

text: i

text: i

text: I

text: i

text: i

text: I

text: Ai

text: (Ai

text: i

text: For

text: every

text: i

text: I,

text: the

text: ith-canonical

text: is

text: the

text: mapping

text: φi

text: Ai

text: Aj

text: i

text: I

text: i

text: I

text: j

text: I

text: deﬁned

text: as

text: φ

text: (x

text: (x

text: i).

text: If

text: no

text: confusions

text: can

text: arise,

text: we

text: can

text: identify

text: A

text: with

text: A

text: i

text: through

text: the

text: i

text: i

text: i

text: i

text: i

text: canonical

text: injection.

text: Therefore,

text: we

text: can

text: see

text: A

text: as

text: a

text: subset

text: of

text: A

text: for

text: every

text: i

text: I.

text: i

text: j

text: j

text: I∈

text: A

text: relation

text: R

text: on

text: a

text: set

text: T

text: is

text: a

text: subset

text: of

text: T

text: T

text: For

text: every

text: x,

text: y

text: T

text: we

text: write

text: xRy

text: to

text: denote

text: (x,

text: y)

text: R.

text: A

text: relation

text: is

text: said

text: to

text: be

text: reﬂexive

text: if

text: xRx

text: for

text: every

text: x

text: T

text: It

text: is

text: symmetric

text: if

text: xRy

text: implies

text: yRx

text: for

text: every

text: x,

text: y

text: T

text: It

text: is

text: anti-symmetric

text: if

text: xRy

text: and

text: yRx

text: imply

text: x

text: y

text: for

text: every

text: x,

text: y

text: T

text: It

text: is

text: transitive

text: if

text: xRy

text: and

text: yRz

text: imply

text: xRz

text: for

text: every

text: x,

text: y,

text: z

text: T

text: An

text: order

text: relation

text: is

text: a

text: relation

text: that

text: is

text: reﬂexive,

text: anti-symmetric

text: and

text: transitive.

text: An

text: equivalence

text: relation

text: is

text: a

text: relation

text: that

text: is

text: reﬂexive,

text: symmetric

text: and

text: transitive.

text: Let

text: R

text: be

text: an

text: equivalence

text: relation

text: on

text: T

text: For

text: every

text: x

text: T

text: the

text: set

text: xˆ

text: y

text: T

text: xRy

text: is

text: the

text: R-equivalence

text: class

text: of

text: x.

text: The

text: collection

text: of

text: R-equivalence

text: classes,

text: which

text: we

text: denote

text: as

text: T/R,

text: forms

text: a

text: partition

text: of

text: T

text: and

text: it

text: is

text: called

text: the

text: quotient

text: space

text: of

text: T

text: by

text: R.

text: The

text: mapping

text: Proj

text: T

text: T/R

text: deﬁned

text: as

text: R

text: Proj

text: (x)

text: xˆ

text: for

text: every

text: x

text: T

text: is

text: the

text: projection

text: mapping

text: onto

text: T/R.

text: R

text: B.

text: Topological

text: spaces

text: U

text: U

text: A

text: topological

text: space

text: is

text: a

text: pair

text: (T,

text: where

text: is

text: a

text: collection

text: of

text: subsets

text: of

text: T

text: satisfying:

text: U

text: U

text: ø

text: and

text: T

text: U

text: U

text: The

text: intersection

text: of

text: a

text: ﬁnite

text: collection

text: of

text: members

text: of

text: is

text: also

text: a

text: member

text: of

text: U

text: U

text: The

text: union

text: of

text: an

text: arbitrary

text: collection

text: of

text: members

text: of

text: is

text: also

text: a

text: member

text: of

text: U

text: U

text: If

text: (T,

text: is

text: a

text: topological

text: space,

text: we

text: say

text: that

text: is

text: a

text: topology

text: on

text: T

text: The

text: power

text: set

text: of

text: T

text: is

text: clearly

text: a

text: topology.

text: It

text: is

text: called

text: the

text: discrete

text: topology

text: on

text: T

text: A

text: A

text: If

text: is

text: a

text: an

text: arbitrary

text: collection

text: of

text: subsets

text: of

text: T

text: we

text: can

text: construct

text: a

text: topology

text: on

text: T

text: starting

text: from

text: as

text: follows:

text: V

text: A⊂V⊂

text: V

text: is

text: a

text: topology

text: on

text: T

text: A

text: A

text: This

text: is

text: the

text: coarsest

text: topology

text: on

text: T

text: that

text: contains

text: It

text: is

text: called

text: the

text: topology

text: on

text: T

text: generated

text: by

text: U

text: U

text: Let

text: (T,

text: be

text: a

text: topological

text: space.

text: The

text: subsets

text: of

text: T

text: that

text: are

text: members

text: of

text: are

text: called

text: the

text: open

text: sets

text: of

text: T

text: Complements

text: of

text: open

text: sets

text: are

text: called

text: closed

text: sets.

text: We

text: can

text: easily

text: see

text: that

text: the

text: closed

text: sets

text: satisfy

text: the

text: following:

text: ø

text: and

text: T

text: are

text: closed.

text: The

text: union

text: of

text: a

text: ﬁnite

text: collection

text: of

text: closed

text: sets

text: is

text: closed.

text: The

text: intersection

text: of

text: an

text: arbitrary

text: collection

text: of

text: closed

text: sets

text: is

text: closed.


text: 3 (cid:92) Let A be an arbitrary subset of T . The closure cl(A) of A is the smallest closed set containing A: cl(A) = F. ⊂ ⊂ A F T, F(cid:91)is closed ◦ The interior A of A is the largest open subset of A: ◦ A = U. ⊂ U A, U is open ⊂ If A T and cl(A) = T , we say that A is dense in T . U (T, ) is said to be separable if there exists a countable subset of T that is dense in T . ∈ ∈ U A subset O of T is said to be a neighborhood of x T if there exists an open set U such that ∈ ⊂ x U O. ∈ O A neighborhood basis of x T is a collection of neighborhoods of x such that for every neighborhood (cid:48) ∈ O (cid:48) ⊂ O of x, there exists O such that O O. U ∈ We say that (T, ) is ﬁrst-countable if every point x T has a countable neighborhood basis. B ⊂ U U ∈ U A collection of open sets is said to be a base for the topology if every open set U can B be written as the union of elements of . U U We say that (T, ) is a second-countable space if the topology has a countable base. It is a well known fact that every second-countable space is ﬁrst-countable and separable. ∈ We say that a sequence (x ) ≥ of elements of T converges to x T if for every neighborhood O ≥ n n 0 ≥ ∈ of x, there exists n 0 such that for every n n , we have x O. We say that x is a limit of the 0 0 n sequence (x ) ≥ . Note that the limit does not need to be unique if there is no constraint on the topology. n n 0 C. Separation axioms U ∈ ∈ U ∈ (T, ) is said to be a T -space if for every x, y T , there exists an open set U such that x U ∈ 1 U and y / U. It is easy to see that (T, ) is T if and only if all singletons are closed. U 1 ∈ (T, ) is said to be a Hausdorff space (or T -space) if for every x, y T , there exist two open sets ∈ U ∈ ∈ ∩ 2 U, V such that x U, y V and U V = ø. U If (T, ) is Hausdorff, the limit of every converging sequence is unique. U ∈ (T, ) is said to be regular if for every x T and every closed set F not containing x, there exist two ∈ U ∈ ⊂ ∩ open sets U, V such that x U, F V and U V = ø. U (T, ) is said to be normal if for every two disjoint closed sets A and B, there exist two open sets ∈ U ⊂ ⊂ ∩ U, V such that A U, B V and U V = ø. U If (T, ) is normal, disjoint closed sets can be separated by disjoint closed neighborhoods. I.e., for (cid:48) ∈ U (cid:48) every two disjoint closed sets A and B, there exist two open sets U, U and two closed sets K, K ⊂ ⊂ ⊂ (cid:48) ⊂ (cid:48) ∩ (cid:48) such that A U K, B U K and K K = ø. U (T, ) is said to be a T -space if it is both T and regular. U 3 1 (T, ) is said to be a T -space if it is both T and normal. 4⇒ ⇒ ⇒ 1 It is easy to see that T T T T . 4 3 2 1 D. Relativization U U If (T, ) is a topological space and A is an arbitrary subset of T , then A inherits a topology from U A (T, ) as follows: U { ∩ ∈ U} = A U : U . A U It is easy to check that is a topology on A. U A U If (T, ) is ﬁrst-countable (respectively second-countable, or Hausdorff), then (A, ) is ﬁrst-countable A (respectively second-countable, or Hausdorff). U U If (T, ) is normal and A is closed, then (A, ) is normal. A

text: Let

text: A

text: be

text: an

text: arbitrary

text: subset

text: of

text: T

text: The

text: closure

text: cl(A)

text: of

text: A

text: is

text: the

text: smallest

text: closed

text: set

text: containing

text: A:

text: cl(A)

text: F.

text: A

text: F

text: T,

text: closed

text: The

text: interior

text: A

text: of

text: A

text: is

text: the

text: largest

text: open

text: subset

text: of

text: A:

text: A

text: U.

text: U

text: A,

text: U

text: is

text: open

text: If

text: A

text: T

text: and

text: cl(A)

text: T

text: we

text: say

text: that

text: A

text: is

text: dense

text: in

text: T

text: U

text: (T,

text: is

text: said

text: to

text: be

text: separable

text: if

text: there

text: exists

text: a

text: countable

text: subset

text: of

text: T

text: that

text: is

text: dense

text: in

text: T

text: U

text: A

text: subset

text: O

text: of

text: T

text: is

text: said

text: to

text: be

text: a

text: neighborhood

text: of

text: x

text: T

text: if

text: there

text: exists

text: an

text: open

text: set

text: U

text: such

text: that

text: x

text: U

text: O.

text: O

text: A

text: neighborhood

text: basis

text: of

text: x

text: T

text: is

text: a

text: collection

text: of

text: neighborhoods

text: of

text: x

text: such

text: that

text: for

text: every

text: neighborhood

text: O

text: O

text: of

text: x,

text: there

text: exists

text: O

text: such

text: that

text: O

text: O.

text: U

text: We

text: say

text: that

text: (T,

text: is

text: ﬁrst-countable

text: if

text: every

text: point

text: x

text: T

text: has

text: a

text: countable

text: neighborhood

text: basis.

text: B

text: U

text: U

text: U

text: A

text: collection

text: of

text: open

text: sets

text: is

text: said

text: to

text: be

text: a

text: base

text: for

text: the

text: topology

text: if

text: every

text: open

text: set

text: U

text: can

text: B

text: be

text: written

text: as

text: the

text: union

text: of

text: elements

text: of

text: U

text: U

text: We

text: say

text: that

text: (T,

text: is

text: a

text: second-countable

text: space

text: if

text: the

text: topology

text: has

text: a

text: countable

text: base.

text: It

text: is

text: a

text: well

text: known

text: fact

text: that

text: every

text: second-countable

text: space

text: is

text: ﬁrst-countable

text: and

text: separable.

text: We

text: say

text: that

text: a

text: sequence

text: (x

text: of

text: elements

text: of

text: T

text: converges

text: to

text: x

text: T

text: if

text: for

text: every

text: neighborhood

text: O

text: n

text: n

text: of

text: x,

text: there

text: exists

text: n

text: such

text: that

text: for

text: every

text: n

text: n

text: we

text: have

text: x

text: O.

text: We

text: say

text: that

text: x

text: is

text: a

text: limit

text: of

text: the

text: n

text: sequence

text: (x

text: Note

text: that

text: the

text: limit

text: does

text: not

text: need

text: to

text: be

text: unique

text: if

text: there

text: is

text: no

text: constraint

text: on

text: the

text: topology.

text: n

text: n

text: C.

text: Separation

text: axioms

text: U

text: U

text: (T,

text: is

text: said

text: to

text: be

text: a

text: T

text: -space

text: if

text: for

text: every

text: x,

text: y

text: T

text: there

text: exists

text: an

text: open

text: set

text: U

text: such

text: that

text: x

text: U

text: U

text: and

text: y

text: U.

text: It

text: is

text: easy

text: to

text: see

text: that

text: (T,

text: is

text: T

text: if

text: and

text: only

text: if

text: all

text: singletons

text: are

text: closed.

text: U

text: (T,

text: is

text: said

text: to

text: be

text: a

text: Hausdorff

text: space

text: (or

text: T

text: -space)

text: if

text: for

text: every

text: x,

text: y

text: T

text: there

text: exist

text: two

text: open

text: sets

text: U

text: U,

text: V

text: such

text: that

text: x

text: U,

text: y

text: V

text: and

text: U

text: V

text: ø.

text: U

text: If

text: (T,

text: is

text: Hausdorff,

text: the

text: limit

text: of

text: every

text: converging

text: sequence

text: is

text: unique.

text: U

text: (T,

text: is

text: said

text: to

text: be

text: regular

text: if

text: for

text: every

text: x

text: T

text: and

text: every

text: closed

text: set

text: F

text: not

text: containing

text: x,

text: there

text: exist

text: two

text: U

text: open

text: sets

text: U,

text: V

text: such

text: that

text: x

text: U,

text: F

text: V

text: and

text: U

text: V

text: ø.

text: U

text: (T,

text: is

text: said

text: to

text: be

text: normal

text: if

text: for

text: every

text: two

text: disjoint

text: closed

text: sets

text: A

text: and

text: B,

text: there

text: exist

text: two

text: open

text: sets

text: U

text: U,

text: V

text: such

text: that

text: A

text: U,

text: B

text: V

text: and

text: U

text: V

text: ø.

text: U

text: If

text: (T,

text: is

text: normal,

text: disjoint

text: closed

text: sets

text: can

text: be

text: separated

text: by

text: disjoint

text: closed

text: neighborhoods.

text: I.e.,

text: for

text: U

text: every

text: two

text: disjoint

text: closed

text: sets

text: A

text: and

text: B,

text: there

text: exist

text: two

text: open

text: sets

text: U,

text: U

text: and

text: two

text: closed

text: sets

text: K,

text: K

text: such

text: that

text: A

text: U

text: K,

text: B

text: U

text: K

text: and

text: K

text: K

text: ø.

text: U

text: (T,

text: is

text: said

text: to

text: be

text: a

text: T

text: -space

text: if

text: it

text: is

text: both

text: T

text: and

text: regular.

text: U

text: (T,

text: is

text: said

text: to

text: be

text: a

text: T

text: -space

text: if

text: it

text: is

text: both

text: T

text: and

text: normal.

text: 4⇒

text: It

text: is

text: easy

text: to

text: see

text: that

text: T

text: T

text: T

text: T

text: D.

text: Relativization

text: U

text: U

text: If

text: (T,

text: is

text: a

text: topological

text: space

text: and

text: A

text: is

text: an

text: arbitrary

text: subset

text: of

text: T

text: then

text: A

text: inherits

text: a

text: topology

text: from

text: U

text: A

text: (T,

text: as

text: follows:

text: U

text: U}

text: A

text: U

text: U

text: A

text: U

text: It

text: is

text: easy

text: to

text: check

text: that

text: is

text: a

text: topology

text: on

text: A.

text: U

text: A

text: U

text: If

text: (T,

text: is

text: ﬁrst-countable

text: (respectively

text: second-countable,

text: or

text: Hausdorff),

text: then

text: (A,

text: is

text: ﬁrst-countable

text: A

text: (respectively

text: second-countable,

text: or

text: Hausdorff).

text: U

text: U

text: If

text: (T,

text: is

text: normal

text: and

text: A

text: is

text: closed,

text: then

text: (A,

text: is

text: normal.

text: A


text: 34 → Let Proj : DMCX Y DMC(Xo)Y be deﬁned as Proj(W ) = Wˆ . The continuity of Proj follows from , , the deﬁnition of the quotient topology. Now let A be a closed subset of DMCX Y. We want to show that Proj(A) is closed. , × Since A is closed in DMCX Y, the set DMCX Y A is closed in (DMCX Y)2. On the other hand, RX(o)Y , , × ∩ , , is closed in (DMCX Y)2 by Lemma 8. Therefore, (DMCX Y A) RX(o)Y is closed in (DMCX Y)2 which , × ∩ (cid:8) , , (cid:9) , is compact, hence (DMCX Y A) RX(o)Y is compact. We have: , , × ∩ (cid:48) ∈ (cid:48) (cid:48) ∈ (DMCX Y A) RX(o)Y = (W, W ) (DMCX Y)2 : W RX(o)YW and W A . , , , , → Now deﬁne the mapping g : (DMCX Y)2 DMCX Y as (cid:0) (cid:1) , , (cid:48) g(W, W ) = W. × ∩ × ∩ Let A := g (DMCX Y (cid:8)A) RX(o)Y . Since g is continuous and s(cid:9)ince (DMCX Y A) RX(o)Y is compact, R , , , , A is also compact. We have: R ∈ ∃ (cid:48) ∈ (cid:48) − A = W DMCX Y : W(cid:0) A, W RX(o)YW (cid:1)= Proj 1(Proj(A)). R , , Since(cid:0) DMCX Y is a metr(cid:1)ic space and since A is compact, Proj−1(Proj(A)) = A is closed in , − R \ \ − R DMCX Y. On the other hand, we have Proj 1 DMC(Xo)Y Proj(A) = DMCX Y Proj 1(Proj(A)), hence − , \ , , \ Proj 1 DMC(Xo)Y Proj(A) is open in DMCX Y, which implies that DMC(Xo)Y Proj(A) is open in , , , DMC(Xo)Y. Therefore, Proj(A) is closed in DMC(Xo)Y. , , APPENDIX C PROOF OF PROPOSITION 4 T let Uˆ be an arbitrary non-empty open subset of (DMC(Xo) , X(o) ) and let Proj be the projection onto ,[m] ,[m] − the RX(o) -equivalence classes. Proj 1(Uˆ) is open in the metric space (DMCX , dX ). Therefore, there ,[m]∈ − ,[m] ,[m] exists W DMCX and (cid:15) > 0 such that Proj 1(Uˆ) contains the open ball of center W and radius (cid:15). ,[m] (cid:48) ∈ (cid:48) (cid:48) We will show that there exists W DMCX such that rank(W ) = m > n and dX (W, W ) < (cid:15). (cid:48) ,[m] ,[m] If rank(W ) = m, take W = W . (cid:88) ∈ { − ∈ } Assume that rank(W ) < m. Let P o ∆ , Im(W ) and W 1 : y Im(W ) be as in Section . { } W [m] RX y Let v ∈ be a collection of m vectors in such that: (cid:88)y y [m] · • P o (y) v = 0. W y ∈ y Im(W) (cid:88) • v = 0. y ∈ \ y [m] Im(W) ∈ • For every y [m], v (x) = 0. y ∈X { } x • The vectors v ∈ are pairwise different. y y [m] Such collection can always be found.  (cid:48) ∈ R Let 0 < δ, δ < 1 and deﬁne P o (cid:48) [m] as follows: W  − ∈ (1 δ)P o (y) if y Im(W ), W P o (cid:48)(y) = δ W − | | otherwise. m Im(W ) (cid:40) ∈ ∈ ∈ (cid:48)− Clearly, P o (cid:48) ∆ and P o (cid:48)(y) > 0 for every y [m]. Now for every y [m], deﬁne W 1 as follows: W [m] W y − − (cid:48) ∈ (cid:48)− (1 δ)W 1 + δπX + δ v if y Im(W ), W 1 = (cid:48) y y y πX + δ v otherwise, y ##LTLine##

text: Let

text: Proj

text: DMCX

text: Y

text: DMC(Xo)Y

text: be

text: deﬁned

text: as

text: Proj(W

text: Wˆ

text: The

text: continuity

text: of

text: Proj

text: follows

text: from

text: the

text: deﬁnition

text: of

text: the

text: quotient

text: topology.

text: Now

text: let

text: A

text: be

text: a

text: closed

text: subset

text: of

text: DMCX

text: Y.

text: We

text: want

text: to

text: show

text: that

text: Proj(A)

text: is

text: closed.

text: Since

text: A

text: is

text: closed

text: in

text: DMCX

text: Y,

text: the

text: set

text: DMCX

text: Y

text: A

text: is

text: closed

text: in

text: (DMCX

text: On

text: the

text: other

text: hand,

text: RX(o)Y

text: is

text: closed

text: in

text: (DMCX

text: by

text: Lemma

text: Therefore,

text: (DMCX

text: Y

text: A)

text: RX(o)Y

text: is

text: closed

text: in

text: (DMCX

text: which

text: is

text: compact,

text: hence

text: (DMCX

text: Y

text: A)

text: RX(o)Y

text: is

text: compact.

text: We

text: have:

text: (DMCX

text: Y

text: A)

text: RX(o)Y

text: (W,

text: W

text: (DMCX

text: W

text: RX(o)YW

text: and

text: W

text: A

text: Now

text: deﬁne

text: the

text: mapping

text: g

text: (DMCX

text: DMCX

text: Y

text: as

text: g(W,

text: W

text: W.

text: Let

text: A

text: g

text: (DMCX

text: Y

text: RX(o)Y

text: Since

text: g

text: is

text: continuous

text: and

text: (DMCX

text: Y

text: A)

text: RX(o)Y

text: is

text: compact,

text: R

text: A

text: is

text: also

text: compact.

text: We

text: have:

text: R

text: A

text: W

text: DMCX

text: Y

text: A,

text: W

text: RX(o)YW

text: Proj

text: R

text: DMCX

text: Y

text: is

text: a

text: space

text: and

text: since

text: A

text: is

text: compact,

text: A

text: is

text: closed

text: in

text: R

text: R

text: DMCX

text: Y.

text: On

text: the

text: other

text: hand,

text: we

text: have

text: Proj

text: DMC(Xo)Y

text: Proj(A)

text: DMCX

text: Y

text: Proj

text: hence

text: Proj

text: DMC(Xo)Y

text: Proj(A)

text: is

text: open

text: in

text: DMCX

text: Y,

text: which

text: implies

text: that

text: DMC(Xo)Y

text: Proj(A)

text: is

text: open

text: in

text: DMC(Xo)Y.

text: Therefore,

text: Proj(A)

text: is

text: closed

text: in

text: DMC(Xo)Y.

text: APPENDIX

text: C

text: PROOF

text: OF

text: PROPOSITION

text: T

text: let

text: Uˆ

text: be

text: an

text: arbitrary

text: non-empty

text: open

text: subset

text: of

text: (DMC(Xo)

text: X(o)

text: and

text: let

text: Proj

text: be

text: the

text: projection

text: onto

text: ,[m]

text: ,[m]

text: the

text: RX(o)

text: -equivalence

text: classes.

text: Proj

text: is

text: open

text: in

text: the

text: metric

text: space

text: (DMCX

text: dX

text: Therefore,

text: there

text: ,[m]∈

text: ,[m]

text: ,[m]

text: exists

text: W

text: DMCX

text: and

text: such

text: that

text: Proj

text: contains

text: the

text: open

text: ball

text: of

text: center

text: W

text: and

text: radius

text: ,[m]

text: We

text: will

text: show

text: that

text: there

text: exists

text: W

text: DMCX

text: such

text: that

text: rank(W

text: m

text: n

text: and

text: dX

text: (W,

text: W

text: ,[m]

text: ,[m]

text: If

text: rank(W

text: m,

text: take

text: W

text: W

text: Assume

text: that

text: rank(W

text: m.

text: Let

text: P

text: o

text: Im(W

text: and

text: W

text: y

text: Im(W

text: be

text: as

text: in

text: Section

text: W

text: [m]

text: RX

text: y

text: Let

text: v

text: be

text: a

text: collection

text: of

text: m

text: vectors

text: in

text: such

text: that:

text: y

text: [m]

text: P

text: o

text: (y)

text: v

text: W

text: y

text: y

text: Im(W)

text: v

text: y

text: y

text: [m]

text: Im(W)

text: For

text: every

text: y

text: [m],

text: v

text: (x)

text: y

text: ∈X

text: x

text: The

text: vectors

text: v

text: are

text: pairwise

text: different.

text: y

text: y

text: [m]

text: Such

text: collection

text: can

text: always

text: be

text: found.

text: R

text: Let

text: δ,

text: δ

text: and

text: deﬁne

text: P

text: o

text: [m]

text: as

text: follows:

text: W

text: δ)P

text: o

text: (y)

text: if

text: y

text: Im(W

text: W

text: P

text: o

text: δ

text: W

text: otherwise.

text: m

text: Im(W

text: Clearly,

text: P

text: o

text: and

text: P

text: o

text: for

text: every

text: y

text: [m].

text: Now

text: for

text: every

text: y

text: [m],

text: deﬁne

text: W

text: as

text: follows:

text: W

text: [m]

text: W

text: y

text: δ)W

text: δπX

text: δ

text: v

text: if

text: y

text: Im(W

text: W

text: y

text: y

text: y

text: πX

text: δ

text: v

text: otherwise,

text: y

text: ##LTLine##


text: 41 → × Now deﬁne the mapping f× : ∆X ∆X ∆X as f×(p) = (p, f(p)), and deﬁne the measure γ × − f on ∆X ∆X as the push-forward of MP by f×, i.e., γ (B) = MP(f×1(B)) for every Borel set B of × f ∆X ∆X . It is easy to see that the m(cid:90)arginals of γ on the ﬁrst an(cid:90)d second factors are MP and MP f f respectively. Therefore, γ is a coupling between MP and MP , hence f f (cid:90) (cid:48) · (cid:48) ≤ (cid:48) · (cid:48) W (MP, MP ) = inf d(p, p ) dγ(p, p ) d(p, p ) dγ (p, p ) 1 f ∈ × × f γ Γ(MP,MPf) ∆X ∆X ∆X ∆X · (≤b) (cid:15) (a) = d(p, f(p)) dMP(p) , 2 ∆X where (a) follows from the fact that γ is the push-forward of MP by f×(p) = (p, f(p)). (b) follows f ∈ from the fact that d(p, f(p)) < (cid:15) for every p ∆X . Therefore, MP well approximates MP and it is 2 f ﬁnitely supported. However, MP may not be balanced, so more work needs to be done in order to ﬁnd f a(cid:90)balanced and ﬁnitely s(cid:90)upported meta-probability measure that well approximates MP. ∈ X (cid:88) (cid:88) For every x , we have: · · n (≥b) n ≥ 1 − (cid:15) (a) p(x) dMP (p) = (f(p))(x) dMP(p) = p (x)MP(A ) l MP(A ) |X | |X |, f i i i,x i 12 ∆X ∆X i=1 i=1 where (a) follo(cid:90)ws from the fact that MP is the push-forward of MP by f. (b) follows from the fact that ∈ ≥ (cid:88)≤f ≤ (cid:88) p A and so p (x) l for every 1 i n. Similarly, we have i i i i,x · n (≤c) n ≤ 1 (cid:15) p(x) dMP (p) = p (x)MP(A ) u MP(A ) |X | + |X |, f i i i,x i 12 ∆X i=1 i=1 ∈ ≤ ≤ ≤ where (c) follows from the fact(cid:12)that p (cid:90)A and so p (x) u(cid:12) for every 1 i n. We conclude that ∈ X (cid:12) i i i (cid:12)i,x for every x , we have (cid:12) (cid:12) (cid:12) (cid:12) − · ≤ (cid:15) πX (x) p(x) dMP (p) |X |, f ∆X(cid:90) 12 X ∈ where πX is the uniform distribution on . Deﬁne p˜ ∆X as: · p˜ = p dMP (p). f ∆X ∈ X For every x , deﬁne − (cid:88) (cid:48) 6(πX (x) p˜(x)) p (x) = + p˜(x). (cid:15) (cid:48) Clearly, p (x) = 1. Moreover, ∈X x (cid:18) − (cid:19) (cid:48) 6(πX (x) p˜(x)) p (x) = + p˜(x) (cid:15) − − (cid:15) 6 πX (x) πX (x) |X | (≥a) 12 1 − (cid:15) 1 − (cid:15) ≥ + |X | |X | = |X | |X | 0, (cid:15) 12 2 12 | − | ≤ (cid:15) (cid:48) ∈ Where (a) follows from the fact that πX (x) p˜(x) |X |. We conclude that p ∆X . Now deﬁne (cid:16) (cid:17) 12 (cid:48) the meta-probability measure MP as follows: (cid:48) (cid:15) · − (cid:15) MP = δ (cid:48) + 1 MP , p f 6 6 (cid:48) where δ (cid:48) is a Dirac measure centered at p . p ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Now

text: deﬁne

text: the

text: mapping

text: f×

text: ∆X

text: ∆X

text: ∆X

text: as

text: f×(p)

text: (p,

text: f(p)),

text: and

text: deﬁne

text: the

text: measure

text: γ

text: f

text: on

text: ∆X

text: ∆X

text: as

text: the

text: push-forward

text: of

text: MP

text: by

text: f×,

text: i.e.,

text: γ

text: (B)

text: for

text: every

text: Borel

text: set

text: B

text: of

text: f

text: ∆X

text: ∆X

text: It

text: is

text: easy

text: to

text: see

text: that

text: the

text: of

text: γ

text: on

text: the

text: ﬁrst

text: second

text: factors

text: are

text: MP

text: and

text: MP

text: f

text: f

text: respectively.

text: Therefore,

text: γ

text: is

text: a

text: coupling

text: between

text: MP

text: and

text: MP

text: hence

text: f

text: f

text: W

text: (MP,

text: MP

text: inf

text: d(p,

text: p

text: dγ(p,

text: p

text: d(p,

text: p

text: dγ

text: (p,

text: p

text: f

text: f

text: γ

text: Γ(MP,MPf)

text: ∆X

text: ∆X

text: ∆X

text: ∆X

text: (≤b)

text: (a)

text: d(p,

text: f(p))

text: dMP(p)

text: ∆X

text: where

text: (a)

text: follows

text: from

text: the

text: fact

text: that

text: γ

text: is

text: the

text: push-forward

text: of

text: MP

text: by

text: f×(p)

text: (p,

text: f(p)).

text: (b)

text: follows

text: f

text: from

text: the

text: fact

text: that

text: d(p,

text: f(p))

text: for

text: every

text: p

text: ∆X

text: Therefore,

text: MP

text: well

text: approximates

text: MP

text: and

text: it

text: is

text: f

text: ﬁnitely

text: supported.

text: However,

text: MP

text: may

text: not

text: be

text: balanced,

text: so

text: more

text: work

text: needs

text: to

text: be

text: done

text: in

text: order

text: to

text: ﬁnd

text: f

text: and

text: ﬁnitely

text: meta-probability

text: measure

text: that

text: well

text: approximates

text: MP.

text: X

text: For

text: every

text: x

text: we

text: have:

text: n

text: (≥b)

text: n

text: (a)

text: p(x)

text: dMP

text: (p)

text: (f(p))(x)

text: dMP(p)

text: p

text: (x)MP(A

text: l

text: MP(A

text: |X

text: |X

text: f

text: i

text: i

text: i,x

text: i

text: ∆X

text: ∆X

text: where

text: (a)

text: from

text: the

text: fact

text: that

text: MP

text: is

text: the

text: push-forward

text: of

text: MP

text: by

text: f.

text: (b)

text: follows

text: from

text: the

text: fact

text: that

text: p

text: A

text: and

text: so

text: p

text: (x)

text: l

text: for

text: every

text: i

text: n.

text: Similarly,

text: we

text: have

text: i

text: i

text: i

text: i,x

text: n

text: (≤c)

text: n

text: p(x)

text: dMP

text: (p)

text: p

text: (x)MP(A

text: u

text: MP(A

text: |X

text: |X

text: f

text: i

text: i

text: i,x

text: i

text: ∆X

text: where

text: (c)

text: follows

text: from

text: the

text: p

text: and

text: so

text: p

text: (x)

text: for

text: every

text: i

text: n.

text: We

text: conclude

text: that

text: X

text: i

text: i

text: i

text: for

text: every

text: x

text: we

text: have

text: πX

text: (x)

text: p(x)

text: dMP

text: (p)

text: |X

text: f

text: X

text: where

text: πX

text: is

text: the

text: uniform

text: distribution

text: on

text: Deﬁne

text: p˜

text: ∆X

text: as:

text: p˜

text: p

text: dMP

text: (p).

text: f

text: ∆X

text: X

text: For

text: every

text: x

text: deﬁne

text: (x)

text: p˜(x))

text: p

text: (x)

text: p˜(x).

text: Clearly,

text: p

text: (x)

text: Moreover,

text: ∈X

text: x

text: (x)

text: p˜(x))

text: p

text: (x)

text: p˜(x)

text: πX

text: (x)

text: πX

text: (x)

text: |X

text: (≥a)

text: |X

text: |X

text: |X

text: |X

text: Where

text: (a)

text: follows

text: from

text: the

text: fact

text: that

text: πX

text: (x)

text: p˜(x)

text: |X

text: We

text: conclude

text: that

text: p

text: ∆X

text: Now

text: deﬁne

text: the

text: meta-probability

text: measure

text: MP

text: as

text: follows:

text: MP

text: δ

text: MP

text: p

text: f

text: where

text: δ

text: is

text: a

text: Dirac

text: measure

text: centered

text: at

text: p

text: p

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6 U (respectively, regular, locally compact, or second-countable) then (T/R, /R) is Hausdorff (respectively, regular, locally compact, or second-countable). K. Metric spaces × → R A metric space is a pair (M, d), where d : M M + satisﬁes: ∈ • d(x, y) = 0 if and only if x = y for every x, y M. ∈ • Symmetry: d(x, y) = d(y, x) for every x, y M. ≤ (cid:115) ∈ • Triangle inequality: d(x, z) d(x, y) + d(y, z) for ever(cid:88)y x, y, z M. If (M, d) is a metric space, we say that d is a metric (or distance) on M. R n − The Euclidean metric on n is deﬁned as d(x, y) = (x y )2, where x = (x ) ≤ ≤ and y = i i i 1 i n i=1 (y ) ≤ ≤ . iR1 i n R n is second countable. Moreover, a subset of n is compact if and only if it is bounded and closed. ∈ For every x M and every (cid:15) > 0, we deﬁne the open ball of center x and radius (cid:15) as: { ∈ } B (x) = y M : d(x, y) < (cid:15) . (cid:15) U The metric topology on M induced by d is the coarsest topology on M which makes d a continuous × d R mapping from M M to +. It is generated by all the open balls. U The metric topology is always T and ﬁrst-countable. Moreover, (M, ) is separable if and only if it 4 d is second-countable. Since every metric space is Hausdorff, we can see that every subset of a compact metric space is closed if and only if it is compact. Every σ-compact metric space is second-countable. For metric spaces, compactness and sequential compactness are equivalent. → A function f : M M from a metric space (M , d ) to a metric space (M , d ) is said to be 1 2 1 1 2 (cid:48) ∈2 uniformly continuous if for every (cid:15) > 0, there exists δ > 0 such that for every x, x M satisfying (cid:48) (cid:48) 1 d (x, x ) < δ we have d (f(x), f(x )) < (cid:15). 1 → 2 If f : M M is a continuous mapping from a compact metric space (M , d ) to an arbitrary metric 1 2 1 1 space (M , d ), then f is uniformly continuous. 2 2 U U A topological space (T, ) is said to be metrizable if there exists a metric d on T such that is the metric topology on T induced by d. The disjoint union of metrizable spaces is always metrizable. The following theorem shows that all separable metrizable spaces are characterized topologically: U Theorem 2. [10] A topological space (T, ) is metrizable and separable if and only if it is Hausdorff, regular and second countable. L. Complete metric spaces ≥ A sequence (x ) ≥ is said to be a Cauchy sequence in (M, d) if for every (cid:15) > 0, there exists n 0 n n 0 ≥ 0 such that for every n , n n we have d(x , x ) < (cid:15). 1 2 0 n1 n2 Every converging sequence is Cauchy, but the converse is not true in general. A metric space is said to be complete if every Cauchy sequence converges in it. A closed subset of a complete space is always complete. A complete subspace of an arbitrary metric space is always closed. Every compact metric space is complete, but the converse is not true in general. For every metric space (M, d), there exists a superspace (M, d) containing M such that: • (M, d) is complete. • M is dense in (M, d). ∈ • d(x, y) = d(x, y) for every x, y M. The space (M, d) is said to be a completion of (M, d). ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: U

text: (respectively,

text: regular,

text: locally

text: compact,

text: or

text: second-countable)

text: then

text: (T/R,

text: /R)

text: is

text: Hausdorff

text: (respectively,

text: regular,

text: locally

text: compact,

text: or

text: second-countable).

text: K.

text: Metric

text: spaces

text: R

text: A

text: metric

text: space

text: is

text: a

text: pair

text: (M,

text: d),

text: where

text: d

text: M

text: M

text: satisﬁes:

text: d(x,

text: y)

text: if

text: and

text: only

text: if

text: x

text: y

text: for

text: every

text: x,

text: y

text: M.

text: Symmetry:

text: d(x,

text: y)

text: d(y,

text: x)

text: for

text: every

text: x,

text: y

text: M.

text: Triangle

text: inequality:

text: d(x,

text: z)

text: d(x,

text: y)

text: d(y,

text: z)

text: for

text: x,

text: y,

text: z

text: M.

text: If

text: (M,

text: d)

text: is

text: a

text: metric

text: space,

text: we

text: say

text: that

text: d

text: is

text: a

text: metric

text: (or

text: distance)

text: on

text: M.

text: R

text: n

text: The

text: Euclidean

text: metric

text: on

text: n

text: is

text: deﬁned

text: as

text: d(x,

text: y)

text: (x

text: y

text: where

text: x

text: (x

text: and

text: y

text: i

text: i

text: i

text: i

text: n

text: (y

text: i

text: n

text: R

text: n

text: is

text: second

text: countable.

text: Moreover,

text: a

text: subset

text: of

text: n

text: is

text: compact

text: if

text: and

text: only

text: if

text: it

text: is

text: bounded

text: and

text: closed.

text: For

text: every

text: x

text: M

text: and

text: every

text: we

text: deﬁne

text: the

text: open

text: ball

text: of

text: center

text: x

text: and

text: radius

text: as:

text: B

text: (x)

text: y

text: M

text: d(x,

text: y)

text: U

text: The

text: metric

text: topology

text: on

text: M

text: induced

text: by

text: d

text: is

text: the

text: coarsest

text: topology

text: on

text: M

text: which

text: makes

text: d

text: a

text: continuous

text: d

text: R

text: mapping

text: from

text: M

text: M

text: to

text: It

text: is

text: generated

text: by

text: all

text: the

text: open

text: balls.

text: U

text: The

text: metric

text: topology

text: is

text: always

text: T

text: and

text: ﬁrst-countable.

text: Moreover,

text: (M,

text: is

text: separable

text: if

text: and

text: only

text: if

text: it

text: d

text: is

text: second-countable.

text: Since

text: every

text: metric

text: space

text: is

text: Hausdorff,

text: we

text: can

text: see

text: that

text: every

text: subset

text: of

text: a

text: compact

text: metric

text: space

text: is

text: closed

text: if

text: and

text: only

text: if

text: it

text: is

text: compact.

text: Every

text: σ-compact

text: metric

text: space

text: is

text: second-countable.

text: For

text: metric

text: spaces,

text: compactness

text: and

text: sequential

text: compactness

text: are

text: equivalent.

text: A

text: function

text: f

text: M

text: M

text: from

text: a

text: metric

text: space

text: (M

text: d

text: to

text: a

text: metric

text: space

text: (M

text: d

text: is

text: said

text: to

text: be

text: ∈2

text: uniformly

text: continuous

text: if

text: for

text: every

text: there

text: exists

text: δ

text: such

text: that

text: for

text: every

text: x,

text: x

text: M

text: satisfying

text: d

text: (x,

text: x

text: δ

text: we

text: have

text: d

text: (f(x),

text: f(x

text: If

text: f

text: M

text: M

text: is

text: a

text: continuous

text: mapping

text: from

text: a

text: compact

text: metric

text: space

text: (M

text: d

text: to

text: an

text: arbitrary

text: metric

text: space

text: (M

text: d

text: then

text: f

text: is

text: uniformly

text: continuous.

text: U

text: U

text: A

text: topological

text: space

text: (T,

text: is

text: said

text: to

text: be

text: metrizable

text: if

text: there

text: exists

text: a

text: metric

text: d

text: on

text: T

text: such

text: that

text: is

text: the

text: metric

text: topology

text: on

text: T

text: induced

text: by

text: d.

text: The

text: disjoint

text: union

text: of

text: metrizable

text: spaces

text: is

text: always

text: metrizable.

text: The

text: following

text: theorem

text: shows

text: that

text: all

text: separable

text: metrizable

text: spaces

text: are

text: characterized

text: topologically:

text: U

text: Theorem

text: A

text: topological

text: space

text: (T,

text: is

text: metrizable

text: and

text: separable

text: if

text: and

text: only

text: if

text: it

text: is

text: Hausdorff,

text: regular

text: and

text: second

text: countable.

text: L.

text: Complete

text: metric

text: spaces

text: A

text: sequence

text: (x

text: is

text: said

text: to

text: be

text: a

text: Cauchy

text: sequence

text: in

text: (M,

text: d)

text: if

text: for

text: every

text: there

text: exists

text: n

text: n

text: n

text: such

text: that

text: for

text: every

text: n

text: n

text: n

text: we

text: have

text: d(x

text: x

text: Every

text: converging

text: sequence

text: is

text: Cauchy,

text: but

text: the

text: converse

text: is

text: not

text: true

text: in

text: general.

text: A

text: metric

text: space

text: is

text: said

text: to

text: be

text: complete

text: if

text: every

text: Cauchy

text: sequence

text: converges

text: in

text: it.

text: A

text: closed

text: subset

text: of

text: a

text: complete

text: space

text: is

text: always

text: complete.

text: A

text: complete

text: subspace

text: of

text: an

text: arbitrary

text: metric

text: space

text: is

text: always

text: closed.

text: Every

text: compact

text: metric

text: space

text: is

text: complete,

text: but

text: the

text: converse

text: is

text: not

text: true

text: in

text: general.

text: For

text: every

text: metric

text: space

text: (M,

text: d),

text: there

text: exists

text: a

text: superspace

text: (M,

text: d)

text: containing

text: M

text: such

text: that:

text: (M,

text: d)

text: is

text: complete.

text: M

text: is

text: dense

text: in

text: (M,

text: d).

text: d(x,

text: y)

text: d(x,

text: y)

text: for

text: every

text: x,

text: y

text: M.

text: The

text: space

text: (M,

text: d)

text: is

text: said

text: to

text: be

text: a

text: completion

text: of

text: (M,

text: d).

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: An algebraic construction of duality functions for the stochastic (1) U (A ) vertex model and its degenerations q n Jeﬀrey Kuan Abstract U A recent paper [KMMO16] introduced the stochastic (A(1)) vertex model. The stochastic S–matrix U q n is related to the R–matrix of the quantum group (A(1)) by a gauge transformation. We will show that q n a certain function D+ intertwines with the transfer matrix and its space reversal. When interpreting µ the transfer matrix as the transition matrix of a discrete–time totally asymmetric particle system on the Z one–dimensional lattice , the function D+ becomes a Markov duality function D which only depends { µ} µ on q and the vertical spin parameters µ . By considering degenerations in the spectral parameter, x the duality results also hold on a ﬁnite lattice with closed boundary conditions, and for a continuous– time degeneration. This duality function had previously appeared in a multi–species ASEP(q, j) process [Kuan16]. The proof here uses that the R–matrix intertwines with the co–product, but does not explicitly use the Yang–Baxter equation. U It will also be shown that the stochastic (A(1)) is a multi–species version of a stochastic vertex q n model studied in [BorPet16, CorPet16]. This will be done by generalizing the fusion process of [CorPet16] and showing that it matches the fusion of [KuReSk81] up to the gauge transformation. We also show, by direct computation, that the multi–species q–Hahn Boson process (which arises at a special value of the spectral parameter) also satisﬁes duality with respect to D , generalizing the 0 single–species result of [Cor15]. Contents 1 Introduction 1 2 Preliminaries 3 2.1 Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1.1 q–notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1.2 Representation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.4 Lumpability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.5 Operator Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Results from [KMMO16] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 The q–Hahn Boson process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Results from [Kuan16] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.5 Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.6 Relationship to previous results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.6.1 The [CGRS15] framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.6.2 Single–species stochastic vertex model from [BorPet16, CorPet16] . . . . . . . . . . 15 3 Further results about S(z) 16 3.1 Additional Degenerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.1.1 At l = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.1.2 At m = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 → ∞ 3.1.3 At z = 0, z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1

text: An

label: title

text: algebraic

text: construction

text: of

text: duality

text: functions

text: for

text: the

text: stochastic

text: U

text: (A

text: vertex

text: model

text: and

text: its

text: degenerations

text: q

text: n

text: Jeﬀrey

label: author

text: Kuan

text: Abstract

text: U

label: abstract

text: A

text: recent

text: paper

text: introduced

text: the

text: stochastic

text: vertex

text: model.

text: The

text: stochastic

text: S–matrix

text: U

text: q

text: n

text: is

text: related

text: to

text: the

text: R–matrix

text: of

text: the

text: quantum

text: group

text: by

text: a

text: gauge

text: transformation.

text: We

text: will

text: show

text: that

text: q

text: n

text: a

text: certain

text: function

text: D+

text: intertwines

text: with

text: the

text: transfer

text: matrix

text: and

text: its

text: space

text: reversal.

text: When

text: interpreting

text: µ

text: the

text: transfer

text: matrix

text: as

text: the

text: transition

text: matrix

text: of

text: a

text: discrete–time

text: totally

text: asymmetric

text: particle

text: system

text: on

text: the

text: Z

text: one–dimensional

text: lattice

text: the

text: function

text: D+

text: becomes

text: a

text: Markov

text: duality

text: function

text: D

text: which

text: only

text: depends

text: µ}

text: µ

text: on

text: q

text: and

text: the

text: vertical

text: spin

text: parameters

text: µ

text: By

text: considering

text: degenerations

text: in

text: the

text: spectral

text: parameter,

text: x

text: the

text: duality

text: results

text: also

text: hold

text: on

text: a

text: ﬁnite

text: lattice

text: with

text: closed

text: boundary

text: conditions,

text: and

text: for

text: a

text: continuous–

text: time

text: degeneration.

text: This

text: duality

text: function

text: had

text: previously

text: appeared

text: in

text: a

text: multi–species

text: ASEP(q,

text: j)

text: process

text: The

text: proof

text: here

text: uses

text: that

text: the

text: R–matrix

text: intertwines

text: with

text: the

text: co–product,

text: but

text: does

text: not

text: explicitly

text: use

text: the

text: Yang–Baxter

text: equation.

text: U

text: It

text: will

text: also

text: be

text: shown

text: that

text: the

text: stochastic

text: is

text: a

text: multi–species

text: version

text: of

text: a

text: stochastic

text: vertex

text: q

text: n

text: model

text: studied

text: in

text: This

text: will

text: be

text: done

text: by

text: generalizing

text: the

text: fusion

text: process

text: of

text: and

text: showing

text: that

text: it

text: matches

text: the

text: fusion

text: of

text: up

text: to

text: the

text: gauge

text: transformation.

text: We

text: also

text: show,

text: by

text: direct

text: computation,

text: that

text: the

text: multi–species

text: q–Hahn

text: Boson

text: process

text: (which

text: arises

text: at

text: a

text: special

text: value

text: of

text: the

text: spectral

text: parameter)

text: also

text: satisﬁes

text: duality

text: with

text: respect

text: to

text: D

text: generalizing

text: the

text: single–species

text: result

text: of

text: Contents

text: Introduction

text: Preliminaries

text: Deﬁnitions

text: q–notation

text: Representation

text: Theory

text: Duality

text: Lumpability

text: Operator

text: Notation

text: Results

text: from

text: The

text: q–Hahn

text: Boson

text: process

text: Results

text: from

text: Fusion

text: Relationship

text: to

text: previous

text: results

text: The

text: framework

text: Single–species

text: stochastic

text: vertex

text: model

text: from

text: Further

text: results

text: about

text: S(z)

text: Additional

text: Degenerations

text: At

text: l

text: At

text: m

text: At

text: z

text: z


text: 3.2 Stochastic fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.3 Stochasticity of S(z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.4 Lumpability of S(z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.5 Analytic Continutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.5.1 The single–species n = 1 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4 Intertwining of transfer matrices 29 4.1 Equivalent expression for duality function . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.2 The one–site case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.3 Extension to L sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.4 Interpretation as a particle system satisfying duality . . . . . . . . . . . . . . . . . . . . . 34 4.4.1 On the inﬁnite line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.4.2 Closed boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.4.3 Continuous–time zero range process . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5 Descriptions of processes 39 − 5.1 At z = ql m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.2 l = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.2.1 Discrete–time process with blocking . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.2.2 Continuous–time zero–range process . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.3 m = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.4 Conjecture for general l, m, z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 6 “Direct” results for multi–species q–Hahn Boson 42 6.1 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 6.2 The Duality Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 6.3 Lumpability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 A Explicit examples 49 A.1 Fusion for l = 1, m = 2, n = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 A.2 S(z) for l = m = 2, n = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 1 Introduction Over the last 25 years, much work has been done investigating interacting particle systems with a prop- erty called stochastic duality (e.g. [BarCor16, BelSch15, BelSch15-2, BelSch16, BorCor13, BorCorSa14, CGRS15, CGRS16, Cor15, GKRV09, GRV10, Kuan15, Kuan16, Ohku16, Sch97, Sch16, SchSan94]). Du- ality has been shown to be useful for asymptotics [BorCorSa14], weak convergence [CST16], and shock dynamics [BelSch16]. The ﬁrst use of duality in interacting particle systems actually goes back even farther, to 1970 [Spit70]. See also Chapter III of [Ligg] for an exposition and references. A more recent direction of research has been to ﬁnd dualities in multi–species versions of some of these systems. This was done in [BelSch15, BelSch15-2, BelSch16, Kuan15, Kuan16]. In these cases, the interacting particle system satisﬁed a Lie algebra symmetry, and the rank of the Lie algebra corresponded to the number of species of particles. Duality has also been discovered in stochastic vertex models (see e.g. [BorCorGor16, BorPet16, CorPet16]). These vertex models enjoy the property of degenerating to a large class of other probabilistic models, including some of the ones above. However, the proofs of these dualities seem to be ad–hoc, in the sense that they required knowing the duality function beforehand, and did not involve constructing the duality function using the algebraic symmetry of the model. Furthermore, there were no known examples of multi–species vertex models satisfying duality. Thus, it is natural to look for a stochastic vertex model such that (1) self–duality holds with respect to a duality function which can be deﬁned with the algebraic symmetry of the vertex model, and (2) in certain degenerations of the vertex model, previous duality results can be recovered, and (3) is a multi–species generalization of an existing single–species model. 2

text: Stochastic

text: fusion

text: Stochasticity

text: of

text: S(z)

text: Lumpability

text: of

text: S(z)

text: Analytic

text: Continutation

text: The

text: single–species

text: n

text: case

text: Intertwining

text: of

text: transfer

text: matrices

text: Equivalent

text: expression

text: for

text: duality

text: function

text: The

text: one–site

text: case

text: Extension

text: to

text: L

text: sites

text: Interpretation

text: as

text: a

text: particle

text: system

text: satisfying

text: duality

text: On

text: the

text: inﬁnite

text: line

text: Closed

text: boundary

text: conditions

text: Continuous–time

text: zero

text: range

text: process

text: Descriptions

text: of

text: processes

text: At

text: z

text: ql

text: m

text: l

text: Discrete–time

text: process

text: with

text: blocking

text: Continuous–time

text: zero–range

text: process

text: m

text: Conjecture

text: for

text: general

text: l,

text: m,

text: z

text: “Direct”

text: results

text: for

text: multi–species

text: q–Hahn

text: Boson

text: Identities

text: The

text: Duality

text: Result

text: Lumpability

text: A

text: Explicit

text: examples

text: Fusion

text: for

text: l

text: m

text: n

text: S(z)

text: for

text: l

text: m

text: n

text: Introduction

text: Over

text: the

text: last

text: years,

text: much

text: work

text: has

text: been

text: done

text: investigating

text: interacting

text: particle

text: systems

text: with

text: a

text: prop-

text: erty

text: called

text: stochastic

text: duality

text: (e.g.

text: Du-

text: ality

text: has

text: been

text: shown

text: to

text: be

text: useful

text: for

text: asymptotics

text: weak

text: convergence

text: and

text: shock

text: dynamics

text: The

text: ﬁrst

text: use

text: of

text: duality

text: in

text: interacting

text: particle

text: systems

text: actually

text: goes

text: back

text: even

text: farther,

text: to

text: See

text: also

text: Chapter

text: III

text: of

text: [Ligg]

text: for

text: an

text: exposition

text: and

text: references.

text: A

text: more

text: recent

text: direction

text: of

text: research

text: has

text: been

text: to

text: ﬁnd

text: dualities

text: in

text: multi–species

text: versions

text: of

text: some

text: of

text: these

text: systems.

text: This

text: was

text: done

text: in

text: In

text: these

text: cases,

text: the

text: interacting

text: particle

text: system

text: satisﬁed

text: a

text: Lie

text: algebra

text: symmetry,

text: and

text: the

text: rank

text: of

text: the

text: Lie

text: algebra

text: corresponded

text: to

text: the

text: number

text: of

text: species

text: of

text: particles.

text: Duality

text: has

text: also

text: been

text: discovered

text: in

text: stochastic

text: vertex

text: models

text: (see

text: e.g.

text: These

text: vertex

text: models

text: enjoy

text: the

text: property

text: of

text: degenerating

text: to

text: a

text: large

text: class

text: of

text: other

text: probabilistic

text: models,

text: including

text: some

text: of

text: the

text: ones

text: above.

text: However,

text: the

text: proofs

text: of

text: these

text: dualities

text: seem

text: to

text: be

text: ad–hoc,

text: in

text: the

text: sense

text: that

text: they

text: required

text: knowing

text: the

text: duality

text: function

text: beforehand,

text: and

text: did

text: not

text: involve

text: constructing

text: the

text: duality

text: function

text: using

text: the

text: algebraic

text: symmetry

text: of

text: the

text: model.

text: Furthermore,

text: there

text: were

text: no

text: known

text: examples

text: of

text: multi–species

text: vertex

text: models

text: satisfying

text: duality.

text: Thus,

text: it

text: is

text: natural

text: to

text: look

text: for

text: a

text: stochastic

text: vertex

text: model

text: such

text: that

text: self–duality

text: holds

text: with

text: respect

text: to

text: a

text: duality

text: function

text: which

text: can

text: be

text: deﬁned

text: with

text: the

text: algebraic

text: symmetry

text: of

text: the

text: vertex

text: model,

text: and

text: in

text: certain

text: degenerations

text: of

text: the

text: vertex

text: model,

text: previous

text: duality

text: results

text: can

text: be

text: recovered,

text: and

text: is

text: a

text: multi–species

text: generalization

text: of

text: an

text: existing

text: single–species

text: model.


text: | ⊗ (cid:105) | (cid:105) where Ω L denotes the vacuum vector Ω on L sites. This transformation was previously used in [Kuan16], using the framework of [CGRS1(cid:89)5]. (cid:89)By Proposit(cid:89)ion 4.(cid:89)2 of [Kuan16], (cid:104)η|Gr|η(cid:105) = const · n+1 L 1 × n q−ηiy+1η[x1,i], (12) i=1 x=1 [ηix]!q i=1 1≤y<x≤L ∈ Z ≤ ≤ ≤ where for η = (η , . . . , η ) n+1 and 1 i j n + 1, set 1 n+1 η = η + . . . + η . [a,b] a b Here, and below, we say that a function c(η, ξ) is constant under particle conservation if it only depends ≤ ≤ on the values of η1 + . . . + ηL+1 and ξ1 + . . . + ξL+1 for 1 i n + 1. The notation const will i i i i denote a constant under particle conservation. By Remark 2.3, if D(η, ξ) is a duality function then so is c(η, ξ)D(η, ξ), as long as particle conservation is satisﬁed. ∈ U ⊗ · · · ⊗ (cid:0) (cid:1) For u (A(1)), deﬁne the operator on V V : q n m1 mL D(u) = Π⊗L ◦ Gr−1 ◦ ∆(L)(u) ◦ Gr−1 ◦ B−1 ⊗L . − Note that the B in this paper is denoted B 2 in [Kuan16]. Then by Proposition 5.1 of [Kuan16], the (cid:104) | | (cid:105) operator D(u ) has an explicit formula, which is that ξ D(u ) η is equal to 0 0 (cid:89) (cid:89) (cid:80) − − const·x=L1[mx − η[x1,n]]!q i=n1 1{mx−η[x1,n+1−i]≥ξ[x1,i]} [m[mxx− η[ηx1[x,1n,+n−1−i]i] −ξ[xξ1[x,1i,]i]]!q]!q q−ξix( y>x 2η[y1,n+1−i]+η[x1,n+1−i]). (13) It is not hard to see that an equivalent ex(cid:20)pression is (cid:21) (cid:89) (cid:89) (cid:80) − − (cid:104)ξ|D(u0)|η(cid:105) = const · L [η1x]!q · · · [ηnx+1]!q n mx ηη[x1x,n−−i] ξ[x1,i] q−ξix( y>x 2η[y1,n+1−i]+η[x1,n+1−i]). x=1 i=1 n+1 i q (14) To see this, note t(cid:89)hat(cid:20)the indicator term 1{mx(cid:21)−η[x1,n+1−i]≥ξ[x1,i]} can(cid:89)be removed, because if its condition does not hold, then the q–binomial term is zero anyway. The other necessary identity is − − − [ηx]! · · · [ηx ]! n mx η[x1,n−i] ξ[x1,i] = [ηx]! · · · [ηx ]! n −[mx η[x1,n−i]]!q . 1 q n+1 q i=1 ηnx+1−i q 1 q n+1 q i=1 [mx η[x1,n+1−i]]!q[ηnx+1−i]!q ∈ N − Note that the expression for D(u ) is still well–deﬁned even when m / . Letting µ = q mx, the 0 x x operator will sometimes be denoted D (u ) to emphasize the dependence on µ. Additionally (see the µ 0(cid:20) (cid:21) proof of Proposition 5.2(b) of [Kuan16])(cid:89), (cid:89) − − mlxi→m∞[η1x]!q · · · [ηnx+1]!q i=n1 mx ηη[x1nx,+n−1−i]i ξ[x1,i] q = i=n1 qξixη[x1,n+!−i] (15) → ∞ so that if one takes all m and assu(cid:89)me(cid:89)s q > 1, then the limit is x (cid:16)(cid:80) (cid:17) (cid:104)ξ|D0|η(cid:105) = L n qξix y≤x 2η[y1,n+1−i] . (16) x=1 i=1 ∈ U Since u (A ) and does not involve f , the operator D(u ) does not involve the spectral param- 0 q n 0 0 eter. The paper [Kuan16] constructs a multi–species version of a process called ASEP(q, j). The single– species case was introduced in [CGRS15], and is itself a generalization of the usual ASEP, in which up to 2j particles can occupy a lattice site. In the homogeneous case when all m = 2j, [Kuan16] shows x that this multi–species ASEP(q, j) has a duality property with respect to the function D (u ). µ 0 14 ##LTLine## ##LTLine## ##LTLine##

text: where

text: Ω

text: L

text: denotes

text: the

text: vacuum

text: vector

text: Ω

text: on

text: L

text: sites.

text: This

text: transformation

text: was

text: previously

text: used

text: in

text: using

text: the

text: framework

text: of

label: equation

text: of

text: const

text: L

text: n

text: [ηix]!q

text: Z

text: where

text: for

text: η

text: (η

text: η

text: and

text: i

text: j

text: n

text: set

text: η

text: η

text: η

text: [a,b]

text: a

text: b

text: Here,

text: and

text: below,

text: we

text: say

text: that

text: a

text: function

text: c(η,

text: ξ)

text: is

text: constant

text: under

text: particle

text: conservation

text: if

text: it

text: only

text: depends

text: on

text: the

text: values

text: of

text: η1

text: and

text: ξ1

text: for

text: i

text: n

text: The

text: notation

text: const

text: will

text: i

text: i

text: i

text: i

text: denote

text: a

text: constant

text: under

text: particle

text: conservation.

text: By

text: Remark

text: if

text: D(η,

text: ξ)

text: is

text: a

text: duality

text: function

text: then

text: so

text: is

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