Non-isolated point

A non-isolated point $x$ of a topological space $X$ is one for which $\{x\}$ is not open, equivalently every neighbourhood of $x$ contains another point of $X$. A homeomorphism preserves this property because it and its inverse preserve open sets, in particular open singletons. In a product space $X\times Y$, a point $(x,y)$ is isolated exactly when both coordinates are isolated: use basic open rectangles and fix the other coordinate. Thus the non-isolated points of $X\times Y$ are $(X^{\prime }\times Y)\cup (X\times Y^{\prime })$, where the primes denote the non-isolated points in the spaces themselves.