# Show a closed subset of a compact set is also compact

## Homework Statement

Show that if E is a closed subset of a compact set F, then E is also compact.

## Homework Equations

I'm pretty sure you refer back to the Heine-Borel theorem to do this.

"A subset of E of Rk is compact iff it is closed and bounded"

## The Attempt at a Solution

We are deal with metric spaces here. It should seem that I need to prove the same thing as in the second half of the Heine-Borel theorem. My textbook is proving Heine-Borel in a confusing way without clear statement/reason steps that I can apply to my problem.

-From the given information, E is in F
-From the given information, every open cover of F has a finite subcover of F
-From the given information, S \ E is open

...not sure where this leave me

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The Heini-Borel theorem only applies to euclidean spaces. Use the definition of compact.

Dick
Homework Helper
You don't need Heine-Borel for this. Take an open cover of E. If you add the open set S/E to that (S is the whole space, right?), then you have an open cover of F. Since F is compact... Can you continue?

I'm with you so far.....you declare O and open cover of E. Then you take the union of O and S \ E and get an open set. This unified open set has to cover all E and all not-E so it has to be an open cover for all of F.

So how does the compactness of F flow back down to E? (Clearly, I have no idea what I'm doing).

Dick