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I'd like to prove that if $p : Y \to X$ is a covering projection, then $Y$ is locally path-connected if $ X$ is.

I've tried a load of different things, but can't get it to work. Any hints / references to proofs online (I haven't been able to find any) would be greatly appreciated. Thanks

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What is the problem? By hypothesis you have a base of path connected sets around every point in $X$ and $p$ is a local homeomorphism. – t.b. Nov 2 '11 at 18:18
Hint: you want to prove something about small neighborhoods in Y and you're given information about some neighborhoods in X. Being a covering projection says something about small enough neighborhoods in Y actually being homeomorphic to neighborhoods in X; just figure out how the quantifiers go and you're done. – Omar Antolín-Camarena Nov 2 '11 at 18:20

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