9
$\begingroup$

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).

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1
9
$\begingroup$

There is a Java applet that displays all 5-element connected posets at http://www1.chapman.edu/~jipsen/gap/posets.html.

$\endgroup$
3
  • $\begingroup$ I don't get it to work. Do you also see just one yellow thing if you try, and don't know what to do with that? $\endgroup$
    – Rasmus
    Oct 21 '10 at 18:09
  • $\begingroup$ It worked o.k. for me. I got the Hasse diagrams of all connected 5-element posets by clicking on "All connected Posets with 5 elements". $\endgroup$ Oct 22 '10 at 2:11
  • 3
    $\begingroup$ For those who have the same problem: askubuntu.com/questions/9279/problems-with-java-applet $\endgroup$
    – Rasmus
    Oct 25 '10 at 17:32
11
$\begingroup$

Peter Jipsen has granted me permission to post static images of his posets.

4-element connected posets (10):

all 4-element connected posets by Peter Jipson

5-element connected posets (44):

all 5-element connected posets by Peter Jipson

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
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1
  • 1
    $\begingroup$ I happened to see this just now. You might be interested in the article I cited in my answer to Finite Hausdorff topological spaces. The reference I gave also gives the number of connected $T_0$ spaces and connected $T_1$ spaces, and from what I can recall, those numbers don't differ all that much (percentage wise) from the "not necessarily connected" numbers I posted. $\endgroup$ Jan 3 '18 at 18:54
4
$\begingroup$

At the online encyclopedia of integer sequences we find, when we type T_0 topologies several hits. Sequence A028856 is the sequence of homeomorphism classes of T_0 topologies, and A028858 has all connected ones (308 topologies of which 235 connected, on 5 points). No explicit list of spaces, though, but some literature references that might help.

$\endgroup$
2
  • $\begingroup$ I think the number 235 refers to 6-point spaces, doesn't it? For the second number in the sequence, 3, should refer to 3-point spaces (on a set with two elements there is only one connected T_0 topology. $\endgroup$
    – Rasmus
    Oct 21 '10 at 17:49
  • $\begingroup$ So, the number I asked for is 44, I guess. $\endgroup$
    – Rasmus
    Oct 21 '10 at 17:52

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