Let $S$ be the set of all possible topologies over the set of positive integers. We can easily show that there is injection from $\mathcal P(\mathbb{N})$ to $S$, and one from $S$ to $\mathcal P(\mathcal P(\mathbb{N}))$, because of the Continum Hypothesis we are sure that there is bijection between $S$ and one of the sets $\mathcal P(\mathbb{N})$ Or $\mathcal P(\mathcal P(\mathbb{N}))$ , so which set is, and can we define this bijective map. In other words what is the cardinality of $S$?
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2The Continuum Hypothesis, even if you assume it is true, tells you nothing about cardinalities between $|\mathcal{P}(\mathbb{N})|$ and $|\mathcal{P}(\mathcal{P}(\mathbb{N}))|$; in particular, it does not imply that there are no cardinalities strictly between these two. – Chris Eagle Nov 17 '22 at 22:19
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2Does this answer your question? What is the cardinality of the set of all topologies on $\mathbb{R}$? (The question is about $\Bbb R$ but the answers are about any infinite set.) – Anne Bauval Nov 17 '22 at 22:27