Tube lemma generalization proof

[radou]

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 $\subseteq$ U x V $\subseteq$ 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 = $\cup U_{i} \times V_{i}$.

Then we cover A x B with basis elements contained in N, so that $A \times B \subseteq \cup U_{i}' \times V_{i}'$. Since A and B are compact, so is A x B, and for this cover, we have a finite subcover, so that $A \times B \subseteq \cup_{i=1}^n U_{i}' \times V_{i}'$.

Now we have the following relation:

$A \times B \subseteq \cup_{i=1}^n U_{i}' \times V_{i}' \subseteq \cup U_{i} \times V_{i} = N$.

Now, I'm not sure if this relation holds:

$$\cup_{i=1}^n (U_{i}' \times V_{i}') \cap (\cup U_{i} \times V_{i}) \subseteq \cup_{i=1}^n (U_{i}' \cap (\cup U_{i})) \times \cup_{i=1}^n (V_{i}' \cap (\cup V_{i})) \subseteq N$$. If it does, then $$U = \cup_{i=1}^n (U_{i}' \cap (\cup U_{i}))$$ and $$V = \cup_{i=1}^n (V_{i}' \cap (\cup V_{i}))$$ are the sets we were looking for.

 

[radou]

If x = (a, b) is in $$(\cup_{i=1}^n (U_{i}' \times V_{i}')) \cap (\cup U_{i} \times V_{i})$$ then a is in Ui, b is in Vi, for some i, and a is in Ui' and b is in Vi'. So, a is in the intersection of Ui and Ui', for some i, and b is in the intersection of Vi and Vi', for some i, i.e. in their unions, so a is in $$(\cup_{i=1}^n (U_{i}' \cap (\cup U_{i}))) \times (\cup_{i=1}^n (V_{i}' \cap (\cup V_{i})))$$.

Any comments?

Edit: just corrected some LaTeX errors and added some brackets, now it should be OK.

 

[radou]

Still no ideas? I'd like to know if this is correct, so I can finalize the exercise section I'm working on.