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question related to perfect maps preserving compactness

Let $Z$ be a compact topological space and let $Y$ be a topological space. Let $f:Y \rightarrow Z$ be a surjective continuous map so that the preimages of points are always compact. Does $Y$ have to be compact?

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    This was [asked and answered recently](http://math.stackexchange.com/questions/90937/question-related-to-perfect-maps-preserving-compactness).2011-12-14

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let $Y=\Big(\coprod_n (0,1/n]\Big)\coprod\{0\}$ and $Z=[0,1]$ with $f$ just the inclusion on the various pieces of $Y$.

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    @james $Y$ is a disjoint union of subspaces of $Z$ and $f$ is just the identity on these pieces. the point of the construction is that each $z\in Z$ has only finitely many preimages.2011-12-15