- #1

- 94

- 0

If [tex]X[/tex] is a finite set of [tex]n[/tex] elements, is there a way to know how many different topologies can [tex]X[/tex] have?

I think it is some combinatorial problem, but not sure.

Thanks for your help.

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter Damidami
- Start date

In summary, the number of different topologies on a finite set X is a subset of P(P(X)), with an upper bound of 2^(2^|X|). However, determining the exact number of topologies on a finite set is a challenging and possibly impossible task, as not every subset of P(X) is a topology on X. Some resources suggest using Stirling numbers to approximate the number of topologies, but it remains an open question.

- #1

- 94

- 0

If [tex]X[/tex] is a finite set of [tex]n[/tex] elements, is there a way to know how many different topologies can [tex]X[/tex] have?

I think it is some combinatorial problem, but not sure.

Thanks for your help.

Physics news on Phys.org

- #2

- #3

- 1,630

- 4

I feel like there is a somewhat pseudo-random pattern.

- #4

- 1,630

- 4

Tinyboss said:You can find more information here.

This is so cool! It is interesting how stirling numbers show up in so many places.

However, like they say there, there doesn't seem to be an easy way of counting the number of topologies on a random set of cardinality n. Since T_0 is a well-behaved topology, it seems somewhat easier.

Does anybody know whether this is an Open Question or?

- #5

- 94

- 0

It seems it turned to be a difficult and interesting question.

Thanks for your replys!

Thanks for your replys!

Share:

- Replies
- 9

- Views
- 1K

- Replies
- 1

- Views
- 573

- Replies
- 21

- Views
- 2K

- Replies
- 7

- Views
- 306

- Replies
- 1

- Views
- 1K

- Replies
- 5

- Views
- 1K

- Replies
- 5

- Views
- 1K

- Replies
- 8

- Views
- 2K

- Replies
- 5

- Views
- 2K

- Replies
- 6

- Views
- 3K