TITLE: Is there a list of all connected T_0-spaces with 5 points?
QUESTION [9 upvotes]: Is there some place (on the internet or elsewhere) where I can find the number and preferably a list of all (isomorphism classes of) finite connected $T_0$-spaces with, say, 5 points?
In know that a $T_0$-topology on a finite set is equivalent to a partial ordering, and wikipedia tells me that there are, up to isomorphism, 63 partially ordered sets with precisely 5 elements. However, I am only interested in connected spaces, and I'd love to have a list (most preferably in terms of Hasse diagrams).

REPLY [11 votes]: Peter Jipsen has granted me permission to post static images of his posets.
4-element connected posets (10):

5-element connected posets (44):

Since the number (139) of distinct 5-element topological spaces (up to homeomorphism) is manageably small
and there don't seem to be any lists posted anywhere, I am posting one here.  The first (second) row lists smallest closed (open) supersets of singletons and the rows that follow list all nontrivial open sets.
1 T_0
    a b c d e
    -----
    a b c d e
    -----
    a b c d e
    ab ac ad ae bc bd be cd ce de
    abc abd abe acd ace ade bcd bce bde cde
    abcd abce abde acde bcde

2 T_0
    ae b c d e
    -----
    a b c d ae
    -----
    a b c d
    ab ac ad ae bc bd cd
    abc abd abe acd ace ade bcd
    abcd abce abde acde

3 T_0
    ae be c d e
    -----
    a b c d abe
    -----
    a b c d
    ab ac ad bc bd cd
    abc abd abe acd bcd
    abcd abce abde

4 T_0
    ae be ce d e
    -----
    a b c d abce
    -----
    a b c d
    ab ac ad bc bd cd
    abc abd acd bcd
    abcd abce

5 connected T_0, poset (4,2)
    ae be ce de e
    -----
    a b c d abcde
    -----
    a b c d
    ab ac ad bc bd cd
    abc abd acd bcd
    abcd

6 T_0
    ade b c d e
    -----
    a b c ad ae
    -----
    a b c
    ab ac ad ae bc
    abc abd abe acd ace ade
    abcd abce abde acde

7 T_0
    ad be c d e
    -----
    a b c ad be
    -----
    a b c
    ab ac ad bc be
    abc abd abe acd bce
    abcd abce abde

8 T_0
    ade be c d e
    -----
    a b c ad abe
    -----
    a b c
    ab ac ad bc
    abc abd abe acd
    abcd abce abde

9 T_0
    ade b c de e
    -----
    a b c ad ade
    -----
    a b c
    ab ac ad bc
    abc abd acd ade
    abcd abde acde

10
    a b c de de
    -----
    a b c de de
    -----
    a b c
    ab ac bc de
    abc ade bde cde
    abde acde bcde

11 T_0
    ad be ce d e
    -----
    a b c ad bce
    -----
    a b c
    ab ac ad bc
    abc abd acd bce
    abcd abce

12 connected T_0, poset (1,9)
    ade be ce d e
    -----
    a b c ad abce
    -----
    a b c
    ab ac ad bc
    abc abd acd
    abcd abce

13 T_0
    ade be c de e
    -----
    a b c ad abde
    -----
    a b c
    ab ac ad bc
    abc abd acd
    abcd abde

14 connected T_0, poset (4,3)
    ade be ce de e
    -----
    a b c ad abcde
    -----
    a b c
    ab ac ad bc
    abc abd acd
    abcd

15 T_0
    ade bde c d e
    -----
    a b c abd abe
    -----
    a b c
    ab ac bc
    abc abd abe
    abcd abce abde

16 connected T_0, poset (2,6)
    ade bd ce d e
    -----
    a b c abd ace
    -----
    a b c
    ab ac bc
    abc abd ace
    abcd abce

17 connected T_0, poset (2,9)
    ade bde ce d e
    -----
    a b c abd abce
    -----
    a b c
    ab ac bc
    abc abd
    abcd abce

18 T_0
    ade bde c de e
    -----
    a b c abd abde
    -----
    a b c
    ab ac bc
    abc abd
    abcd abde

19
    ade b c de de
    -----
    a b c ade ade
    -----
    a b c
    ab ac bc
    abc ade
    abde acde

20 connected T_0, poset (4,6)
    ade bde ce de e
    -----
    a b c abd abcde
    -----
    a b c
    ab ac bc
    abc abd
    abcd

21 connected T_0, poset (3,6)
    ade bde cde d e
    -----
    a b c abcd abce
    -----
    a b c
    ab ac bc
    abc
    abcd abce

22 connected T_0, poset (5,4)
    ade bde cde de e
    -----
    a b c abcd abcde
    -----
    a b c
    ab ac bc
    abc
    abcd

23
    ade bde c de de
    -----
    a b c abde abde
    -----
    a b c
    ab ac bc
    abc
    abde

24 connected
    ade bde cde de de
    -----
    a b c abcde abcde
    -----
    a b c
    ab ac bc
    abc

25 T_0
    acde b c d e
    -----
    a b ac ad ae
    -----
    a b
    ab ac ad ae
    abc abd abe acd ace ade
    abcd abce abde acde

26 T_0
    acd be c d e
    -----
    a b ac ad be
    -----
    a b
    ab ac ad be
    abc abd abe acd
    abcd abce abde

27 connected T_0, poset (1,2)
    acde be c d e
    -----
    a b ac ad abe
    -----
    a b
    ab ac ad
    abc abd abe acd
    abcd abce abde

28 T_0
    acde b ce d e
    -----
    a b ac ad ace
    -----
    a b
    ab ac ad
    abc abd acd ace
    abcd abce acde

29 connected T_0, poset (1,4)
    ace bde c d e
    -----
    a b ac bd abe
    -----
    a b
    ab ac bd
    abc abd abe
    abcd abce abde

30 connected T_0, poset (2,1)
    acde be ce d e
    -----
    a b ac ad abce
    -----
    a b
    ab ac ad
    abc abd acd
    abcd abce

31 T_0
    acde b ce de e
    -----
    a b ac ad acde
    -----
    a b
    ab ac ad
    abc abd acd
    abcd acde

32 T_0
    ace bd ce d e
    -----
    a b ac bd ace
    -----
    a b
    ab ac bd
    abc abd ace
    abcd abce

33
    ac b c de de
    -----
    a b ac de de
    -----
    a b
    ab ac de
    abc ade bde
    abde acde

34 connected T_0, poset (4,4)
    acde be ce de e
    -----
    a b ac ad abcde
    -----
    a b
    ab ac ad
    abc abd acd
    abcd

35 connected T_0, poset (2,3)
    ace bde ce d e
    -----
    a b ac bd abce
    -----
    a b
    ab ac bd
    abc abd
    abcd abce

36 connected T_0, poset (4,8)
    ace bde ce de e
    -----
    a b ac bd abcde
    -----
    a b
    ab ac bd
    abc abd
    abcd

37 connected T_0, poset (1,5)
    acde bde c d e
    -----
    a b ac abd abe
    -----
    a b
    ab ac
    abc abd abe
    abcd abce abde

38 T_0
    acde b cde d e
    -----
    a b ac acd ace
    -----
    a b
    ab ac
    abc acd ace
    abcd abce acde

39
    a b cde cde e
    -----
    a b cd cd cde
    -----
    a b
    ab cd
    acd bcd cde
    abcd acde bcde

40 connected T_0, poset (2,7)
    acde bd ce d e
    -----
    a b ac abd ace
    -----
    a b
    ab ac
    abc abd ace
    abcd abce

41
    ac bc c de de
    -----
    a b abc de de
    -----
    a b
    ab de
    abc ade bde
    abde

42 connected T_0, poset (3,2)
    acde bde ce d e
    -----
    a b ac abd abce
    -----
    a b
    ab ac
    abc abd
    abcd abce

43 connected T_0, poset (2,4)
    acde bde c de e
    -----
    a b ac abd abde
    -----
    a b
    ab ac
    abc abd
    abcd abde

44 connected T_0, poset (3,1)
    acde be cde d e
    -----
    a b ac acd abce
    -----
    a b
    ab ac
    abc acd
    abcd abce

45 T_0
    acde b cde de e
    -----
    a b ac acd acde
    -----
    a b
    ab ac
    abc acd
    abcd acde

46
    acde b c de de
    -----
    a b ac ade ade
    -----
    a b
    ab ac
    abc ade
    abde acde

47
    ae b cde cde e
    -----
    a b cd cd acde
    -----
    a b
    ab cd
    acd bcd
    abcd acde

48 connected T_0, poset (4,7)
    acde bde ce de e
    -----
    a b ac abd abcde
    -----
    a b
    ab ac
    abc abd
    abcd

49 connected T_0, poset (4,9)
    acde be cde de e
    -----
    a b ac acd abcde
    -----
    a b
    ab ac
    abc acd
    abcd

50
    ac bde c de de
    -----
    a b ac bde bde
    -----
    a b
    ab ac
    abc bde
    abde

51 connected
    ae be cde cde e
    -----
    a b cd cd abcde
    -----
    a b
    ab cd
    acd bcd
    abcd

52 connected T_0, poset (3,7)
    acde bde cde d e
    -----
    a b ac abcd abce
    -----
    a b
    ab ac
    abc
    abcd abce

53 connected T_0, poset (5,5)
    acde bde cde de e
    -----
    a b ac abcd abcde
    -----
    a b
    ab ac
    abc
    abcd

54 connected
    acde bde c de de
    -----
    a b ac abde abde
    -----
    a b
    ab ac
    abc
    abde

55
    acde b cde de de
    -----
    a b ac acde acde
    -----
    a b
    ab ac
    abc
    acde

56 connected
    acde bde cde de de
    -----
    a b ac abcde abcde
    -----
    a b
    ab ac
    abc

57 connected T_0, poset (1,7)
    acde bcde c d e
    -----
    a b abc abd abe
    -----
    a b
    ab
    abc abd abe
    abcd abce abde

58 connected T_0, poset (3,4)
    acde bcde ce d e
    -----
    a b abc abd abce
    -----
    a b
    ab
    abc abd
    abcd abce

59 connected T_0, poset (5,2)
    acde bcde ce de e
    -----
    a b abc abd abcde
    -----
    a b
    ab
    abc abd
    abcd

60 connected
    acde bc c de de
    -----
    a b abc ade ade
    -----
    a b
    ab
    abc ade
    abde

61 connected T_0, poset (3,9)
    acde bcde cde d e
    -----
    a b abc abcd abce
    -----
    a b
    ab
    abc
    abcd abce

62
    acde b cde cde e
    -----
    a b acd acd acde
    -----
    a b
    ab
    acd
    abcd acde

63
    a b cde cde cde
    -----
    a b cde cde cde
    -----
    a b
    ab
    cde
    acde bcde

64 connected T_0, poset (5,7)
    acde bcde cde de e
    -----
    a b abc abcd abcde
    -----
    a b
    ab
    abc
    abcd

65 connected
    acde bcde c de de
    -----
    a b abc abde abde
    -----
    a b
    ab
    abc
    abde

66 connected
    acde be cde cde e
    -----
    a b acd acd abcde
    -----
    a b
    ab
    acd
    abcd

67 connected
    acde bcde cde de de
    -----
    a b abc abcde abcde
    -----
    a b
    ab
    abc

68 connected
    acde bcde cde cde e
    -----
    a b abcd abcd abcde
    -----
    a b
    ab
    abcd

69
    acde b cde cde cde
    -----
    a b acde acde acde
    -----
    a b
    ab
    acde

70 connected
    acde bcde cde cde cde
    -----
    a b abcde abcde abcde
    -----
    a b
    ab

71 connected T_0, poset (1,1)
    abcde b c d e
    -----
    a ab ac ad ae
    -----
    a
    ab ac ad ae
    abc abd abe acd ace ade
    abcd abce abde acde

72 connected T_0, poset (1,3)
    abcde be c d e
    -----
    a ab ac ad abe
    -----
    a
    ab ac ad
    abc abd abe acd
    abcd abce abde

73 connected T_0, poset (2,2)
    abcde be ce d e
    -----
    a ab ac ad abce
    -----
    a
    ab ac ad
    abc abd acd
    abcd abce

74 connected T_0, poset (4,5)
    abcde be ce de e
    -----
    a ab ac ad abcde
    -----
    a
    ab ac ad
    abc abd acd
    abcd

75
    abc b c de de
    -----
    a ab ac de de
    -----
    a
    ab ac de
    abc ade
    abde acde

76 connected T_0, poset (1,6)
    abcde bde c d e
    -----
    a ab ac abd abe
    -----
    a
    ab ac
    abc abd abe
    abcd abce abde

77 connected T_0, poset (2,8)
    abcde bd ce d e
    -----
    a ab ac abd ace
    -----
    a
    ab ac
    abc abd ace
    abcd abce

78 connected T_0, poset (3,3)
    abcde bde ce d e
    -----
    a ab ac abd abce
    -----
    a
    ab ac
    abc abd
    abcd abce

79 connected T_0, poset (2,5)
    abcde bde c de e
    -----
    a ab ac abd abde
    -----
    a
    ab ac
    abc abd
    abcd abde

80 connected
    abcde b c de de
    -----
    a ab ac ade ade
    -----
    a
    ab ac
    abc ade
    abde acde

81
    ab b cde cde e
    -----
    a ab cd cd cde
    -----
    a
    ab cd
    acd cde
    abcd acde

82 connected T_0, poset (5,1)
    abcde bde ce de e
    -----
    a ab ac abd abcde
    -----
    a
    ab ac
    abc abd
    abcd

83
    abc bc c de de
    -----
    a ab abc de de
    -----
    a
    ab de
    abc ade
    abde

84
    a bc bc de de
    -----
    a bc bc de de
    -----
    a
    bc de
    abc ade
    bcde

85 connected T_0, poset (3,8)
    abcde bde cde d e
    -----
    a ab ac abcd abce
    -----
    a
    ab ac
    abc
    abcd abce

86 connected
    abe b cde cde e
    -----
    a ab cd cd acde
    -----
    a
    ab cd
    acd
    abcd acde

87 connected T_0, poset (5,6)
    abcde bde cde de e
    -----
    a ab ac abcd abcde
    -----
    a
    ab ac
    abc
    abcd

88 connected
    abcde bde c de de
    -----
    a ab ac abde abde
    -----
    a
    ab ac
    abc
    abde

89 connected
    abe be cde cde e
    -----
    a ab cd cd abcde
    -----
    a
    ab cd
    acd
    abcd

90 connected
    abcde bde cde de de
    -----
    a ab ac abcde abcde
    -----
    a
    ab ac
    abc

91 connected T_0, poset (1,8)
    abcde bcde c d e
    -----
    a ab abc abd abe
    -----
    a
    ab
    abc abd abe
    abcd abce abde

92
    a bcde bcde d e
    -----
    a bc bc bcd bce
    -----
    a
    bc
    abc bcd bce
    abcd abce bcde

93 connected T_0, poset (3,5)
    abcde bcde ce d e
    -----
    a ab abc abd abce
    -----
    a
    ab
    abc abd
    abcd abce

94 connected
    ae bcde bcde d e
    -----
    a bc bc bcd abce
    -----
    a
    bc
    abc bcd
    abcd abce

95
    a bcde bcde de e
    -----
    a bc bc bcd bcde
    -----
    a
    bc
    abc bcd
    abcd bcde

96 connected T_0, poset (5,3)
    abcde bcde ce de e
    -----
    a ab abc abd abcde
    -----
    a
    ab
    abc abd
    abcd

97 connected
    abcde bc c de de
    -----
    a ab abc ade ade
    -----
    a
    ab
    abc ade
    abde

98 connected
    ae bcde bcde de e
    -----
    a bc bc bcd abcde
    -----
    a
    bc
    abc bcd
    abcd

99
    ade bc bc de de
    -----
    a bc bc ade ade
    -----
    a
    bc
    abc ade

100 connected T_0, poset (4,1)
    abcde bcde cde d e
    -----
    a ab abc abcd abce
    -----
    a
    ab
    abc
    abcd abce

101 connected
    abcde b cde cde e
    -----
    a ab acd acd acde
    -----
    a
    ab
    acd
    abcd acde

102 connected
    ade bcde bcde d e
    -----
    a bc bc abcd abce
    -----
    a
    bc
    abc
    abcd abce

103 connected T_0, poset (5,8)
    abcde bcde cde de e
    -----
    a ab abc abcd abcde
    -----
    a
    ab
    abc
    abcd

104 connected
    abcde bcde c de de
    -----
    a ab abc abde abde
    -----
    a
    ab
    abc
    abde

105 connected
    abcde be cde cde e
    -----
    a ab acd acd abcde
    -----
    a
    ab
    acd
    abcd

106
    ab b cde cde cde
    -----
    a ab cde cde cde
    -----
    a
    ab
    cde
    acde

107 connected
    ade bcde bcde de e
    -----
    a bc bc abcd abcde
    -----
    a
    bc
    abc
    abcd

108
    a bcde bcde de de
    -----
    a bc bc bcde bcde
    -----
    a
    bc
    abc
    bcde

109 connected
    abcde bcde cde de de
    -----
    a ab abc abcde abcde
    -----
    a
    ab
    abc

110 connected
    ade bcde bcde de de
    -----
    a bc bc abcde abcde
    -----
    a
    bc
    abc

111 connected
    abcde bcde cde cde e
    -----
    a ab abcd abcd abcde
    -----
    a
    ab
    abcd

112 connected
    abcde b cde cde cde
    -----
    a ab acde acde acde
    -----
    a
    ab
    acde

113 connected
    abcde bcde cde cde cde
    -----
    a ab abcde abcde abcde
    -----
    a
    ab

114 connected
    abcde bc bc de de
    -----
    a abc abc ade ade
    -----
    a
    abc ade

115 connected
    abcde bcde bcde d e
    -----
    a abc abc abcd abce
    -----
    a
    abc
    abcd abce

116
    a bcde bcde bcde e
    -----
    a bcd bcd bcd bcde
    -----
    a
    bcd
    abcd bcde

117 connected
    abcde bcde bcde de e
    -----
    a abc abc abcd abcde
    -----
    a
    abc
    abcd

118 connected
    ae bcde bcde bcde e
    -----
    a bcd bcd bcd abcde
    -----
    a
    bcd
    abcd

119 connected
    abcde bcde bcde de de
    -----
    a abc abc abcde abcde
    -----
    a
    abc

120 connected
    abcde bcde bcde bcde e
    -----
    a abcd abcd abcd abcde
    -----
    a
    abcd

121
    a bcde bcde bcde bcde
    -----
    a bcde bcde bcde bcde
    -----
    a
    bcde

122 connected
    abcde bcde bcde bcde bcde
    -----
    a abcde abcde abcde abcde
    -----
    a

123
    abc abc c de de
    -----
    ab ab abc de de
    -----
    ab de
    abc
    abde

124 connected
    abe abe cde cde e
    -----
    ab ab cd cd abcde
    -----
    ab cd
    abcd

125 connected
    abcde abcde c d e
    -----
    ab ab abc abd abe
    -----
    ab
    abc abd abe
    abcd abce abde

126 connected
    abcde abcde ce d e
    -----
    ab ab abc abd abce
    -----
    ab
    abc abd
    abcd abce

127 connected
    abcde abcde ce de e
    -----
    ab ab abc abd abcde
    -----
    ab
    abc abd
    abcd

128 connected
    abcde abcde cde d e
    -----
    ab ab abc abcd abce
    -----
    ab
    abc
    abcd abce

129 connected
    abcde abcde cde de e
    -----
    ab ab abc abcd abcde
    -----
    ab
    abc
    abcd

130 connected
    abcde abcde c de de
    -----
    ab ab abc abde abde
    -----
    ab
    abc
    abde

131 connected
    abcde abcde cde de de
    -----
    ab ab abc abcde abcde
    -----
    ab
    abc

132
    ab ab cde cde cde
    -----
    ab ab cde cde cde
    -----
    ab
    cde

133 connected
    abcde abcde cde cde e
    -----
    ab ab abcd abcd abcde
    -----
    ab
    abcd

134 connected
    abcde abcde cde cde cde
    -----
    ab ab abcde abcde abcde
    -----
    ab

135 connected
    abcde abcde abcde d e
    -----
    abc abc abc abcd abce
    -----
    abc
    abcd abce

136 connected
    abcde abcde abcde de e
    -----
    abc abc abc abcd abcde
    -----
    abc
    abcd

137 connected
    abcde abcde abcde de de
    -----
    abc abc abc abcde abcde
    -----
    abc

138 connected
    abcde abcde abcde abcde e
    -----
    abcd abcd abcd abcd abcde
    -----
    abcd

139 connected
    abcde abcde abcde abcde abcde
    -----
    abcde abcde abcde abcde abcde
    -----
    trivial topology