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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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I also asked this at math.stackexchange.com/questions/7295/… but did not receive much response. –  Rasmus Bentmann Oct 21 '10 at 17:01

2 Answers 2

up vote 4 down vote accepted

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

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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? –  Rasmus Bentmann 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". –  Richard Stanley Oct 22 '10 at 2:11
    
Ok, thank you for the information. –  Rasmus Bentmann Oct 22 '10 at 13:41
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For those who have the same problem: askubuntu.com/questions/9279/problems-with-java-applet –  Rasmus Bentmann Oct 25 '10 at 17:32

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.

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

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