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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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marked as duplicate by Dylan Moreland, Matt N., Thomas Andrews, Chris Eagle, Jonas Teuwen Dec 14 '11 at 23:33

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

This was asked and answered recently. –  Dylan Moreland Dec 14 '11 at 23:25

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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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what do you mean by an "inclusion on the various pieces of Y" –  james Dec 15 '11 at 1:54
@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. –  yoyo Dec 15 '11 at 2:10

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