Compact subset of a locally compact space

[ehrenfest]

Homework Statement

How would I prove that if X is locally compact and a subset of X, V, is compact, then there is an open set G with $$V \subset G$$ and closure(G) compact?EDIT: X is also Hausdorff (which with local compactness implies that it is regular) if that matters

Homework Equations

The Attempt at a Solution

 

[morphism]

If X is regular, then given a point x in X and a nbhd* U of x, we can find a nbhd W of x whose closure sits in U. Do this for each x in V. Now use the facts that X is locally compact and that V is compact.

(*: I'm using nbhd to mean an open set containing x.)

 

[ehrenfest]

So because X is locally compact, we have a nbhd around every point x whose closure is compact, call it N_x. Now, like you said, for all x in V, we take find a nbhd around x, call it M_x whose closure sits in N_x.

The union of M_x over x in V form an open cover of V, so we have a finite subcover, say {M_x_1 to M_x_n}. Union M_x_i is an open set that contains V. Is its closure compact, though?

 

[ehrenfest]

Okay. I see why the closure is compact. But also, how would I prove that the boundary of our open set G is compact?

 

[morphism]

It's a closed subset of a compact space, isn't it?

 

[ehrenfest]

Got it. Thanks.