TITLE: What does "open set" mean in the concept of a topology?
QUESTION [24 upvotes]: Given the following definition of topology, I am confused about the concept of "open sets".

2.2 Topological Space. We use some of the properties of open sets in the case of metric spaces in order to define what is meant in general by a class of open sets and by a topology.
Definition 2.2. Let $X$ be a nonempty set. A topology $\mathcal{T}$ for $X$ is a collection of subsets of $X$ such that $\emptyset,X\in\mathcal{T}$, and $\mathcal{T}$ is closed under arbitrary unions and finite intersections.
    We say $(X,\mathcal{T})$ is a topological space. Members of $\mathcal{T}$ are called open sets.
    If $x\in X$ then a neighbourhood of $x$ is an open set containing $x$.

It seems to me that the definition of an open subset is that subset $A$ of a metric space $X$ is called open if for every point $x \in A$ there exists $r>0$ such that $B_r(x)\subseteq A$. What is the difference of being open in a metric space and being open in a topological space?
Thanks so much.

REPLY [3 votes]: Here is an interesting intuitive definition for open set that actually gives rise to the topological axioms!

An open set is a collection of objects for which we can easily verify membership of any given member.

Note that it may not be easy to verify non-membership of non-members!


Now let us take a look at the axioms:

The empty collection is an open set.

Vacuously true since it has no members.

Any finite intersection of open sets is an open set.

True because you just verify membership in each open set one at a time.

Any union of open sets is an open set.

True because for each member in the union you just have to verify its membership in the correct open set of the union.


Notice how the restriction to "finite intersection" is crucial here because we may not be able to easily verify membership in infinitely many open sets! Here we are using the intuitive assumption that it is easy to do finitely many easy tasks.
Now of course one can ask whether there is a concrete representation of such intuitive notion of open sets. There is! In a metric space, suppose that you interpret that membership of $x$ in $S$ is easily verified iff there is some positive error margin $ε$ such that moving $x$ by a distance of at most $ε$ cannot make it no longer a member of $S$. Then indeed you can easily verify that metric-open sets have easily verifiable membership, and you can see why infinite intersections of metric-open sets may not be open (since those sets may require smaller and smaller error margins for that member).

Incidentally, there is a related notion of semi-decidable sets (namely the sets such that there is a program that accepts the input iff it is a member) or equivalently recursively enumerable (RE) sets (namely the sets such that a program can enumerate all their members by appending them one by one to the end of the designated output list). RE sets are slightly related to open classes in linguistics, in that the enumeration idea corresponds to the notion of a collection whose members are not fixed but get added over time but never removed.
We have the following analogous facts for RE sets:

The empty set is an RE set.

It is enumerated by a program that rejects everything.

Any finite intersection of RE sets is an RE set.

Take any RE sets $S_{1..n}$, and let $P_{1..n}$ be the corresponding programs. Then $\bigcap_{k \in \{1..n\}} S_k$ is enumerated by a program that runs all the programs $P_{1..n}$ in parallel and appends an object to the output list whenever it has been appended to all of their output lists. (Again note how the finiteness restriction for intersections naturally arises from the fact that we can only run finitely many programs in parallel!)

Any RE union of RE sets is an RE set.

(Note that an RE set is represented by a program, so by "RE union of RE sets" we mean "union of RE sets whose programs form an RE set".) Take any set $S$ of RE sets such that there is a program $P$ that enumerates all the programs corresponding to the RE sets in $S$. Then $\bigcup S$ is enumerated by a program that runs $P$ and in parallel runs every program output by $P$ (which we shall call a child program) and appends an object to the output list whenever it is appended for the first time to the output list for some child program.