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Set Theory/Compactness Question

  1. Apr 12, 2012 #1
    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 Ux, Vx, open in X,Y respectively such that x[itex]\in[/itex]Ux, L[itex]\subseteq[/itex] Vx and Ux x Vx [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 Ux over x in X, and V as the union of Vx 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 (Ux x Vx) 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 Vx 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
     
  2. jcsd
  3. Apr 12, 2012 #2

    micromass

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    The idea is indeed to take the intersection of the [itex]V_x[/itex], 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 [itex]\{U_x~\vert~x\in K\}[/itex] is an open cover of K.
     
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