**Just to recap:** Let $M$ be any set. A set $\mathcal{O}\_M \subseteq P(M)$ is a topology on $M$ if it satisfies the three things that it satisfies (I won't write them down since it's not relevant to my question).

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My question is directed toward the fact that $\mathcal{O}\_M \subseteq P(M)$.

We know that for any set $M, |P(M)| = 2^{|M|}.$

Does this imply that all topologies always have a cardinality of $2^n$ for some $n \in \mathbb{N}$?