Solution: Let $a\in A$ be fixed. For any $b \in B$ the point $(a,b) \in A \times B$ and $N$ is an open neighbourhood of $A$ implies, there exists open sets $U_b(a) \subseteq X$ and $V_b(a) \subseteq Y$ such that \[ (a,b) \in U_b(a) \times V_b(a) \subseteq N. \] The collection $\mathcal{V} \coloneqq \left\{ V_b(a): b \in B \right\} $ of open sets is an open cover of $B$ and $B$ is compact implies $\mathcal{V} $ has a finite subcover of $B$, say, $\left\{ V_{b_i}(a) : i = 1,2,\dots, k \right\} $ such that $B \subseteq \cup _{i=1}^k V_{b_i}(a)$. Let \[ U(a) = \bigcap_{i=1}^{k} U_{b_i}(a) \text{ and } V(a) = \bigcup_{i=1}^{k} V_{b_i}(a). \] It is clear that the sets $U(a)$ and $V(a)$ are open subsets of $X$ and $Y$ respectively. We also have \[ \mathcal{U} \coloneqq \left\{ U(a): a \in A \right\} \] is an open cover of $A$ and as $A$is compact it posses a finite subcover, say \[ \left\{ U(a_i) : i = 1,2, \dots, l \right\}. \]

Set \[ U = \bigcup_{i=1}^{l} U(a_i) \text{ and } V = \bigcap_{i=1}^{l} V(a_i). \] Now it is clear that $U$ and $V$ are open subsets of $X$ and $Y$ respectively and \[ A \times B \subseteq U \times V \subseteq N. \]