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Suppose that X and Y are compact. Let F be an open cover for XxY. Then, for y in Y, F is an open cover for Xx{y}, which is compact. Hence F has a finite subcover

Fy = [tex]\{ U^{y}_{1}\times V^{y}_{1},...,U^{y}_{n}\times V^{y}_{n} \} [/itex] where y is in all [tex]V^{y}_{i}[/itex]

This is the step I don't get. Why is y in all those sets? I understand y must be in at least one (to be covered) but why all? The proof uses this fact to construct a non-empty set so it's pretty crucial, but for the life of me I don't understand how to deduce it :(

I was thinking maybe it's because of the axiom of choice, but I don't know much about that to even be sure it involves it.