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Homework Statement
Let A and B be compact subspaces of X and Y, respectively. Let N be an open set in X x Y containing A x B. One needs to show that there exist open sets U in X and V in Y such that A x B [itex]\subseteq[/itex] U x V [itex]\subseteq[/itex] N.
The Attempt at a Solution
Here's my try:
First of all, since N is open, it can be written as a union of basis elements in X x Y, i.e. let N = [itex]\cup U_{i} \times V_{i}[/itex].
Then we cover A x B with basis elements contained in N, so that [itex]A \times B \subseteq \cup U_{i}' \times V_{i}'[/itex]. Since A and B are compact, so is A x B, and for this cover, we have a finite subcover, so that [itex]A \times B \subseteq \cup_{i=1}^n U_{i}' \times V_{i}'[/itex].
Now we have the following relation:
[itex]A \times B \subseteq \cup_{i=1}^n U_{i}' \times V_{i}' \subseteq \cup U_{i} \times V_{i} = N[/itex].
Now, I'm not sure if this relation holds:
[tex]\cup_{i=1}^n (U_{i}' \times V_{i}') \cap (\cup U_{i} \times V_{i}) \subseteq \cup_{i=1}^n (U_{i}' \cap (\cup U_{i})) \times \cup_{i=1}^n (V_{i}' \cap (\cup V_{i})) \subseteq N[/tex]. If it does, then [tex]U = \cup_{i=1}^n (U_{i}' \cap (\cup U_{i}))[/tex] and [tex]V = \cup_{i=1}^n (V_{i}' \cap (\cup V_{i}))[/tex] are the sets we were looking for.
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