(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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 [itex]\subseteq[/itex] 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[itex]\in[/itex]U_{x}, L[itex]\subseteq[/itex] V_{x}and U_{x}x V_{x}[itex]\subseteq[/itex] W.

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

2. Relevant 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 [itex]\subseteq[/itex] 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

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# Homework Help: Set Theory/Compactness Question

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