19

I would like to know if there is like a magical formula to know how many topologies exist on a finite set

For example for $X = \{ a, b, c \}$ I found $29$, but I dont know if there are more or how to know this exact number without writing all topologies first.

gnometorule
  • 4,640
Gmath
  • 412
  • 2
    See https://oeis.org/A000798 and the references therein. – Robert Israel Feb 17 '13 at 20:37
  • This link maybe helpful for you: http://www.jstor.org/stable/2313548 – Anirban Oct 24 '14 at 12:40
  • 3
    In case you are interested and didn't know your question is equivalent to: how many preorders exists on a finite set. For any topological space $(X,\tau)$ you can define $x\leq y$ if and only $x \in U \Rightarrow y \in U$ giving a preorder $(X,\leq)$. Conversely given a preorder $(X,\leq)$ one can define a topology on $X$ by setting $U \subseteq X$ open if $x \in U \wedge x \leq y \Rightarrow y \in U$ (i.e. $U$ is up closed). Restricting to finite topological spaces and finite preorders we find that these maps are inverse to each other. – Nex Nov 03 '15 at 10:33
  • 2
    Another Q is how many inequivalent topologies there are on a finite set. (Equivalent meaning homeomorphic.) I dk whether a formula for this has been found. – DanielWainfleet Jan 20 '17 at 21:46
  • Have a look here: https://arxiv.org/pdf/1503.08359.pdf and here: https://cs.uwaterloo.ca/journals/JIS/VOL9/Benoumhani/benoumhani11.pdf – SARTHAK GUPTA Mar 03 '20 at 15:19

1 Answers1

6

The number 29 is apparently correct. See oeis. It is the same as the number of quasi orders on the set, as well as some other equivalent formulations. I don't know any quick way to compute it; maybe because I can tell a doughnut from a coffee cup. ..

All kidding aside (or almost all ), consider a set with $5$ labeled elements. Then there are $2^5$, or $32$ subsets to "work with "...

Now, we need to see how many ways some of these can be selected which meet the definition of a topology...

Namely, $\emptyset $ and the whole set, call it $X $; and closure under finite (which is automatic, here) intersections, and arbitrary unions...

There are, amazing in a way, $6942$ ways to do this...

If you think of how many ways you can choose subsets, you're looking at the power set again, so $2^{2^5} $ ways... Now, $2^{10}\approx 10^3$, so we're looking at on the order of $10^9$ collections that we can use to try to build a topology. ..

Only about $1$ in every $10^6$ subsets of the power set qualifies as a topology. ..

The moral of the story seems to be that we have our work cut out for us... and remember we are only at $n=5$...

Community
  • 1