= 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 $(X'\times Y)\cup(X\times Y')$, where the primes denote the non-isolated points in the spaces themselves.
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