What is an example of a topological space $(X,\tau)$ with the properties that


1. $X\cong X\setminus \{x\}$ for all $x\in X$, and
2. $(X,\tau)$ is not topologically [homogeneous](https://en.wikipedia.org/wiki/Homogeneous_space)


?



Non-homogeneous space $X$ such that $X\cong X\setminus \{x\}$ for all $x\in X$
An infinite disjoint union of half-open and open intervals works.


