Set Theory/Compactness Question

[Zoe-b]

Homework Statement

This is part of a question I've managed to do most of.. I've got as far as:

Let K, L be compact subsets of topological spaces X,Y respectively, and let K x L \subseteq W where W is open in X x Y.

I have already shown that: for each x in K there exist sets U_x, V_x, open in X,Y respectively such that x\inU_x, L\subseteq V_x and U_x x V_x \subseteq W.

Now I have to show there exist sets U, V open in X,Y respectively and such that K x L \subseteq U x V \subseteq W.

Homework Equations

I'm pretty stuck- If I set U as the union of U_x over x in X, and V as the union of V_x over x in X, I obviously get that K x L \subseteq U x V, but I don't think this in necessarily contained in W. Alternatively if I take a union of (U_x x V_x) over x in X then this IS contained in W but can't be written in the form U x V (I don't think). I also haven't yet used the compactness property- this must be relevant but not really sure where..

I also considered taking V as the intersection of the V_x over x but this isn't necessarily open.

Any clues welcome! I have mock exams soon, need to get my head around this stuff :/

Thanks

 

[micromass]

The idea is indeed to take the intersection of the V_x, but this isn't open: an arbitrary intersection of open sets isn't necessarily open. BUT a finite intersection of open sets IS open. So we must reduce this collection to a finite collection somehow.

To do this, notice that \{U_x~\vert~x\in K\} is an open cover of K.