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.