Are there topological spaces $X,Y$, each having more than $2$ points, such that

• $X\not\cong Y$, 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

?

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.