Topology — the tube lemma

Let $X$ and $Y$ be topological spaces. Consider the following statement:

S: For every open subset $U\subseteq X\times Y$ and every $x\in X$ such that $\{x\}\times Y\subseteq U$, there is a neighbourhood $W$ of $x$ in $X$ such that $W\times Y\subseteq U$.

Which of the following statements is true?