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## Homework Statement

Let [itex] p: E \rightarrow B [/itex] be a covering map.

If B is compact and[itex]p^{-1}(b)[/itex] is finite for each b in B, then E compact.

Note: This is a problem from Munkres pg 341, question 6b in section 54.

## The Attempt at a Solution

I begin with a cover of E denote it [itex]\{U_\alpha\}[/itex].

I want to reduce this to a finite subcover (thus showing that E is compact).

First I use the fact that p is a covering map and thus open to send this cover of E to a cover of B.

Denote the image of [itex]\{U_\alpha\}[/itex] under p by [itex]\{W_\alpha\}[/itex]

Then since B is compact I can reduce this to a finite subcover: [itex]\cup_{i=1}^n W_i[/itex].

Here is where I get stuck, I'm not sure how to send this finite subcover of B back over to E. I'm not even sure if I'm going about this the right way.

Any help is greatly appreciated, thanks.

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