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Is there a non empty set $X$ such that $|T_X| = 2^{2^{|X|}}$ where $T_X$ are all topologies on $X$?

I know $T_X$ to be a strict subset of $\mathcal{P}(\mathcal{P}(X))$ and that no finite set works.

Logarithmnepnep
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    The empty set does not work. The only topology on the empty set is ${\varnothing}$. But $P(\varnothing)={\varnothing}$, and $P(P(\varnothing))={\varnothing,{\varnothing}}$, so there is no bijection. – Arturo Magidin Oct 09 '23 at 16:19
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    https://math.stackexchange.com/questions/65731/what-is-the-cardinality-of-the-set-of-all-topologies-on-mathbbr – Jakobian Oct 09 '23 at 16:26
  • The answer in the question linked-to by Jakobian answers your question: any infinite set. – Arturo Magidin Oct 09 '23 at 16:27

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