Is the every open set in a product space contains an open set with compact boundry?

Question

Let N be an open set of YxZ. Is there exist an open set W in YxZ with compact boundry such that W is a subset of N?

Answer 1

Not in general, no. For example, let $Y$ be an infinite set with the particular point topology, and $Z$ be a one-point space. Then $N=\{(p,*)\}$ (where $p$ is the particular point of $Y$ and $*$ is the only point of $Z$ is a counterexample.