# Compact Topological Spaces [duplicate]

Possible Duplicate:
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?

-

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

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 $Y$.
@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