Is there some way to find the cardinality of set of all clopen subsets of a topological space, say, Cantor space, Baire space?
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I don’t have an answer to the general question, but I can answer it for the specific spaces mentioned. The clopen algebra of the Cantor space is the free Boolean algebra on $\omega$ generators, which has cardinality $\omega$. For $n\in\omega$ let $B_n=\{n\}\times\omega^\omega$. Then $\bigcup_{n\in A}B_n$ is a clopen subset of $\omega^\omega$ for each $A\subseteq\omega$, so the Baire space has at least $2^\omega$ clopen sets. On the other hand, $\omega^\omega$ is second countable, so it has only $2^\omega$ open sets and therefore precisely $2^\omega$ clopen sets. |
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