If $N$ open in $X\times Y$, $Y$ compact, containing $\{x_0\}\times Y$ then exists open $W\subseteq X$ containing $x_0$, s.t $W\times Y\subseteq N$ 
Let $X,Y$ be topological spaces, $Y$ compact space. Let $x_0\in X$. Prove that if $N$ is an open set in $X\times Y$ containing $\{x_0\}\times Y$ then there is an open set $W\subseteq X$ containing $x_0$ such that $W\times Y\subseteq N$

Suppose $N\subseteq X\times Y$
Then $N=\bigcup_{\alpha\in J}(B_1\times B_2)_\alpha$ where $B_1,B_2$ are basic sets in $X,Y$. Such that $\{x_0\}\times Y\subseteq N$.
then $x_0\in B_{1_\alpha}$ for some basic set. And since $\{x_0\}\times Y\subseteq N$, then $\bigcup_{\alpha\in J} B_{2_\alpha}$ covers $Y$.
I'm not sure how to do this since I know unions don't distribute nicely with cartesian products. 
So I don't think I can just say that $B_{1_\alpha}\times \bigcup_{\alpha\in J} B_{2_\alpha}\subseteq N$. 
I also don't see where compactness of $Y$ matters.
 A: 
I don't see where compactness of $Y$ matters.

Consider $N = \{(x,y) \in \mathbb{R}\times\mathbb{R} \mid y < e^{-x}\}$.  Then $\mathbb{R}\times \{0\} \subset N$ but there is no open set $W \subset \mathbb{R}$ such that $0 \in W$ and $\mathbb{R} \times W \subset N$ (because $\displaystyle\lim_{x\to\infty} e^{-x} = 0$).  The problem, of course, is that $\mathbb{R}$ is not compact.

As for your original question, first define $J_0 = \{\alpha \in J \mid x_0 \in B_{1,\alpha}\}$.  From here, we have $\{x_0\} \times Y \subseteq \bigcup_{\alpha \in J_0}B_{1,\alpha}\times B_{2,\alpha}.$
Since $Y \subseteq \bigcup_{\alpha \in J_0}B_{2,\alpha}$ is an open cover of the compact $Y$, there is a finite $I \subset J_0$ such that $Y \subseteq \bigcup_{\alpha \in I} B_{2,\alpha}$
Since $x_0 \in B_{1,\alpha}$ for all $\alpha \in J_0 \supseteq I,$ and $I$ is finite, it follows that $x_0 \in \bigcap_{\alpha \in I} B_{1,\alpha},$ which is a finite intersection of open sets and therefore open.
Now define $W = \bigcap_{\alpha \in I} B_{1,\alpha},$ and check that $W \times Y \subseteq N$
