We say that a topological space $(X,\tau)$ is *point-removal insensitive* if for all $x\in X$ we have $X\cong X\setminus \{x\}$.


If $X,Y$ are point-removal insensitive, does this imply that $X\times Y$ with the product topology is point-removal insensitive?



Product of point-removal insensitive spaces
Let $X = {\mathbb R} \setminus {\mathbb Z}$, that is, $X$ is the free union of countably many open intervals. Then $X$ is point-removable insensitive because the removal of any point leaves a free union of countably many open intervals. However, $X \times X$ is not point-removal insensitive because every component of $X \times X$ just a product of intervals and is, therefore, simply connected, but if any point is removed from $X \times X$, that point's component is not simply connected.


