Is it known (in ZFC) that every connected compact Hausdorff space has cardinality at least $\mathfrak c=|\mathbb R|$?
2026-03-27 23:40:18.1774654818
Cardinality of a connected compact Hausdorff space?
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Yes, assuming the space has more than one point. A compact Hausdorff space $X$ is normal, so by Urysohn's lemma given any two distinct points $x,y\in X$ there is a continuous map $f:X\to[0,1]$ such that $f(x)=0$ and $f(y)=1$. If $X$ is connected, then $f$ must be surjective, and so in particular $X$ has cardinality at least $\mathfrak{c}$.