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I've seen on that there is currently no known formula for the amount of distinct topologies on a finite set. However I was wondering there are some rough estimates, aside from the trivial ones on the amount.

Given a finite set $X$ of cardinality $n$ we have the trivial upper bound of $2^2^n$. I was wondering whether there are some stronger estimates.
In particular, I was wondering whether one can show there are less topologies on $X$ then there are relations on $Xtimes X$? i.e:

Can we say that it is larger than $2^n^2$ or smaller than $2^n^2$?

I would also welcome any help in estimates for an infinite set $X$.

See The number of finite topologies by Kleitman and Rothschild. If $T_n$ is the number of topologies on a set with $n$ elements, they show that
$$log_2 T_nsim n^2/4$$
This is quite an old paper and there may be a better asymptotic formula at this point. Asymptotically, the logarithm of the number of topologies is also equal to the logarithm of the number of partial orders on the set. The same authors have a sequel to the paper that gives a better formula for the number of partial orders, but without the logarithm I believe they cease to be asymptotically equal.

On a finite set $S$, the topology is uniquely determined by the closures of singletons: A subset of $S$ is closed iff it contains the closure of all its elements.
Given a topology, we can define
$$tag1ale biff overlineasubseteq overlineb$$
(or equivalently, $ain overlineb$).
This need not be a partial order because we can have $ale b$ and $ble a$ with $ane b$. But it is a reflexive and transitive relation on $S$.

On the other hand, from a a reflexive and transitive relation $le$ on $S$, we can define a topology by declaring
$$tag2 Atext closediffforall ain Acolon forall xin Scolon xle ato xin A.$$
One verifies that the union or intersection of two closed sets is closed and trivially $emptyset$ and $S$ are closed.

Moreover, the associations topology $leftrightarrow$ reflexive transitive relation in $(1)$ and $(2)$ are inverse of each other, hence

On a finite set, there are as many topologies as there are reflexive transitive binary relations.

The relation to partial orders is that a reflexive transitive binary relation on $S$ is the same as a partial order on a partition of $S$. In particular, if the counts of topologies and partial orders are $T_n$ and $P_n$, respectively, we have
$$ T_n=P_n+nchoose 2P_n-1+left(nchoose 3+frac12nchoose 2n-2choose 2right)P_n-2+ldots+2^n-1P_2+P_1.$$