# Non-homogeneous space $X$ such that $X\cong X\setminus \{x\}$ for all $x\in X$

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

?

An infinite disjoint union of half-open and open intervals works.

• Don't you need infinitely many half-open and infinitely many open intervals? – Matthias Wendt Nov 9 '17 at 11:50

Take the disjoint union of any two nonhomeomorphic spaces with that property as long as they are perfect, e.g., $\mathbb{Q}\coprod(\mathbb{R}-\mathbb{Q})$.