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Let $X$ be a connected topological space, and $\pi : Y \rightarrow X$ a surjective covering space map. Suppose that the group of deck transformations of $\pi$ contains a subgroup $\mathbb Z_p$, where $p$ is a prime number, such that $\mathbb Z_p$ acts freely and transitively on fibers of $\pi$. If $Y$ is not connected, is it necessarily that $Y$ is homeomorphic to a disjoint union of $p$ copies of $X$? (Note: Here we do not assume that $X$ is locally connected.)

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    By "a subgroup $\mathbb{Z}_p$", do you mean "a subgroup which is isomorphic to $\mathbb{Z}_p$"? And I assume in this context $\mathbb{Z}_p$ means the cyclic group of order p?2012-08-17
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    @HarryAltman Yes. I should write more clearly.2012-08-17
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    Does anyone have examples of pathological covering spaces, perhaps with X not locally connected? That might be instructive.2012-08-24

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