Are there topological spaces $X,Y$, each having more than $2$ points, satisfying the following two properties?


* $X\not\cong Y$, and
* there is a bijection $\varphi: X\to Y$ such that for all $x\in X$ the spaces $X\setminus \{x\}$ and $Y\setminus \{\varphi(x)\}$ are homeomorphic.



Non-homeomorphic spaces such that taking away a point makes them homeomorphic
If $X$ and $Y$ are Hausdorff then the one easily sees that the homeomorphism can not be induced by $\varphi$. But you did not request that.


Let $C\_0$ denote a disjoint union of countably many copies of the Cantor set. It is easily seen that the Cantor set $C$ is homeomorphic to the one point compactification of $C\_0$. Since $C$ is homogeneous, by removing any point from $C$ we get $C\_0$. Then $C\_0$ less any point is again homeomorphic to $C\_0$. Thus take $X\cong C$, $Y\cong C\_0$ and $\varphi$ to be any bijection between them.


