I read somewhere that for $\mathbb{R}$, the number of topologies is indeed the same as the obvious upper bound, namely $2^{2^{|\mathbb{R}|}}$. Does the same hold for all infinite sets $X$, i.e. is the cardinality of the set of topologies on $X$ equal to $2^{2^{|X|}}$? How would one prove it?
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1Does this answer your question? What is the cardinality of the set of all topologies on $\mathbb{R}$? (The accepted answer seems to be what you are after.) – Feb 26 '21 at 17:32