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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

?

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2 Answers 2

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

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    $\begingroup$ Don't you need infinitely many half-open and infinitely many open intervals? $\endgroup$
    – Matthias Wendt
    Commented Nov 9, 2017 at 11:50
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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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