TITLE: Questions about existence of injections between infinite sets and the sets of all infinite topologies on them
QUESTION [5 upvotes]: 1) If $X$ is an infinite set and $T_X$ the set of all infinite topologies on $X$ is it in general true that there is no injection $f_T:T_X \to X$?
2) What conditions on $X$ assure an injection (if that´s ever possible)?
By infinite topologies, I mean topologies with an infinite number of sets (subsets of $X$) as its elements.

REPLY [2 votes]: For an infinite set $X$, there are $2^{2^{|X|}}$ topologies with infinitely many elements on $X$.
The upper bound is obvious, the lower bound is given by my comment (and the slight correction in YCor's): for any ultrafilter $U$ on $X$, define $T_U$ to have as opens the empty set and the elements of $U$. It is clear that this forms a topology, and that you can recover $U$ from it. 
Moreover, clearly $T_U$ has infinitely many open sets. 
So $U\mapsto T_U$ defines an injection $\beta X\to T_X$, and it's well known that $|\beta X| = 2^{2^{|X|}}$.
So there can be no injection $T_X\to X$ (in fact, no injection $T_X\to P(X)$)
YCor's comment below mine elaborates a bit on what one can ask of these topologies and how many we get (Hausdorff, metrizable)