Let $X$ be a compact connected Hausdorff space and $f:X\rightarrow X$ a continuous open map. Show that f is onto.
2026-03-31 10:43:26.1774953806
Show that f is onto.
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Since $X$ is compact, $f(X)$ must be compact, hence closed, because $X$ is a Hausdorff space. But $f$ is an open map, hence $f(X)$ must be open, too. Since in a connected space the only open and closed sets are the empty set and the space itself, the map $f$ must be surjective.