Let

$$ \forall_{A\subseteq X}\quad \mathcal B_A\ :=\ \{{a} :\ a\in A\}\cup X $$

Then each $\ \mathcal B_A\ $ is a topological base in $\ X\ $ while every two of them generate different topologies in $\ X.\ $ When $\ X\ $ is infinite then

$$ |\,\{\mathcal B_A : \, A\subset X\ \ \text{and}\ \ |A|=\infty \}\,| \,\ =\,\ 2^{|X|} $$

We see that an injection of the set of all topologies in $\ X\ $ into $\ X\ $ is impossible when $\ X\ $ infinite.

Remark 1   There is no such injection also when $\ X\ $ is finite but $\ |X|\ne 1,\ $ i.e. there is never any injection of the set of all topologies in $\ X\ $ into $\ X\ $ except for $\ X\ $ being a singleton.

Remark 2   The situation, cardinality wise, is even more dramatic.

Let

$$ \forall_{A\subseteq X}\quad \mathcal B_A \ :=\ \{\{a\} :\ a\in A\}\cup X $$

Then each $\ \mathcal B_A\ $ is a topological base in $\ X\ $ while every two of them generate different topologies in $\ X.\ $ When $\ X\ $ is infinite then

$$ |\,\{\mathcal B_A : \, A\subset X\ \ \text{and}\ \ |A|=\infty \}\,| \,\ =\,\ 2^{|X|} $$

We see that an injection of the set of all topologies in $\ X\ $ into $\ X\ $ is impossible when $\ X\ $ infinite.

Remark 1   There is no such injection also when $\ X\ $ is finite but $\ |X|\ne 1.\ $ Indeed, let

$$ \forall_{A\subseteq X}\quad T_A \ :=\ \{\emptyset\,\ A\,\ X\} $$

These are topologies, and

$$ |\,\{T_A:\,A\subseteq X\}\,|\ =\ 2^{|X|}-1 $$

for every non-empty finite $\, X.\, $ In the empty case we get $\, 0\, $ points and $\, 1\, $ topology, $\ 0<1.$

Remark 2   The situation, cardinality wise, is even more dramatic.