TITLE: How many topologies can be defined on a finite set with $n$ element?
QUESTION [9 upvotes]: we know that a topology on $A$ is a subset of the power set of $A$, with two properties;
$1)$ it is closed under finite intersection and arbitrary union.
$2)$ it must include $A$ and the empty set.
Now,if $A$ is a finite set with $n$ elements, how many topologies can be defined on $A$?  
Or, how many topologies can be defined on $A$ up to a homeomorphism?

REPLY [3 votes]: This problem seems nontrivial. Any papers I've found on the topic suggest that this is still an open problem. See the following question on MathOverflow:
You might also see the following paper:
The Number of Topologies on a Finite Set

Or this paper:
Enumeration of Finite Topologies