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**Problem:**

Suppose that $X$ and $Y$ are each compact sets. Consider $X \times Y$.

Let $g_x = \left\{x\right\} \times Y$ and $G = \left\{g_x : x \in X\right\}$.

Show that if $g_x \in G$ and $U \subset X \times Y$ open with $g_x \subset U$, then $\exists V$ open in $X \times Y$ such that:

(i) $g_x \subset V$

(ii) for any $g_x' \in G$, if $g_x' \cap V \neq \emptyset$ then $g_x' \subset U$

(meaning, $G$ is an upper semi-continuous collection)

If anyone has any hints to give I would appreciate it!

We are looking at vertical strips.

I considered $g_x \in G$, some $x \in X$, and an open set $U$ containing $g_x$. Of course I need to construct a $V$ that satisfies these conditions. Any hints? I thought maybe to use compactness, as $X \times Y$ is compact and every open cover has a finite subcover.