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.

  • 1
    $\begingroup$ Don't you need infinitely many half-open and infinitely many open intervals? $\endgroup$ – 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., $\mathbb{Q}\coprod(\mathbb{R}-\mathbb{Q})$.


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