From this Clopen subsets of a compact metric space we know that any compact metric space $X$ contains at most countably many clopen subsets ; my question is : Does there exist a compact metric space $X$ containing countably infinitely many clopen subsets ?
2026-04-24 13:56:45.1777039005
Does there exist a compact metric space $X$ containing countably infinitely many clopen subsets?
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Consider the subset of $\mathbb R$ $$\{ \frac{1}{n} | n\in \mathbb N^*\} \cup \{0\}\;$$ which inherits a metric structure from the standard metric of $\mathbb R$. The sets $\{1/n\}$ are countably infinitely many clopen subsets.