# etale space v. covering space

On the Wikipedia page Sheaf, under the section "The etale space of a sheaf," the author claims that the etale space of the sheaf of (continuous) sections of a continuous map $Y \to X$ is (homeomorphic to) $Y$ if and only if $Y \to X$ is a covering map. This doesn't seem quite right to me. Isn't the appropriate condition weaker? I think that it should be something like a local homeomorphism with discrete fibers, unless I misunderstood the Wikipedia article. Am I correct?

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You are right. The condition is that $Y\to X$ is an etale map, i.e. a local homeomorphism. (it is the standard equivalence between sheaves and etale maps)
So covering map $\Leftrightarrow$ étale map? This is a surprise to me. I think my attempts to locally trivialize an arbitrary étale map rely on tacit assumptions. – Eivind Dahl Jun 13 '11 at 10:59
@Eivin Dahl: no, there are many more etale maps that coverings (typically, $Y$ is very non-Hausdorff in the etale maps corresponding to sheaves). I just believe the statement in the Wikipedia article is wrong; in place of covering it should say etale. – user8268 Jun 13 '11 at 12:43