Give an example of topological spaces $X$ and $Y$ and a map $f: X \to Y$ that $f$ is an open, closed, and continuous but not a local homeomorphism. (The map and the topological spaces have to satisfy these 4 characteristics simultaneously)
2026-03-26 02:58:15.1774493895
$f$ Open, Closed and continuous but not a local homeomorphism
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Let $f: [0,1] \times [0,1] \to [0,1]$, all in the usual topology, be given by $f(x,y)=x$ (the projection).
$f$ is open and continuous, as all projections are.
$f$ is closed as the domain is compact and the codomain is Hausdorff.
$f$ is not a local homeomorphism for dimension reasons, basically. Or: all open sets in the domain cannot be disconnected by removing a point, while (basic) open sets in the codomain can be, so they can never be homeomorphic.