Covering Spaces : Compact Hausdorff Spaces and Homomorphisms

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Covering Spaces: Algebraic Topology problem
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Assume X and Y are arcwise connected and locally arcwise connected,
X is compact Hausdorff, and Y is Hausdorff. Let f: X-->Y be a local homeomorphism. Prove that (X,f) is a covering space.

Solution:

an arc connecting two points and of a topological space is not simply (like a path) a continuous function such that and , but must also have a continuous inverse function, i.e., that it is a homeomorphism between and the image of f.

If s is compact subspace in a ...