Let $X$ be a compact Hausdorff space and $C(X)$ be the set of continuous functions on $X$. And $F$ is a closed subspace of $X$. If the $f\in C(X)$ such that $f|_{F}=0$ is only zero function( i.e. $f=0$), then $F=X$???
2026-04-06 03:07:27.1775444847
A question about compact Hausdorff space
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Yes, by Urysohn's Lemma : If $F \neq X$, then there is some $x\in X\setminus F$, and a continuous function $f$ such that $f(x) = 1$, and $f = 0$ on $F$.