# Is there a list of all connected T_0-spaces with 5 points?

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).

## 3 Answers

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

• 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? Oct 21 '10 at 18:09
• 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". Oct 22 '10 at 2:11
• For those who have the same problem: askubuntu.com/questions/9279/problems-with-java-applet Oct 25 '10 at 17:32

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

• 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. Jan 3 '18 at 18:54

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.

• 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. Oct 21 '10 at 17:49
• So, the number I asked for is 44, I guess. Oct 21 '10 at 17:52