(Theorem) Tube lemma

Theorem: Tube lemma

If $Y$ is compact and $N\subseteq X\times Y$ is an open set containing $\{x_0\}\times Y$ for some $x_0\in X$, then there exists a neighborhood $W\subseteq X$ of $x_0$ that

$$\{x_0\}\times Y\subseteq W\times Y\subseteq N.$$