What is an example of a topological space $(X,\tau)$ with the properties that
- $X\cong X\setminus \{x\}$ for all $x\in X$, and
- $(X,\tau)$ is not topologically homogeneous
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What is an example of a topological space $(X,\tau)$ with the properties that
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An infinite disjoint union of half-open and open intervals works.
Take the disjoint union of any two nonhomeomorphic spaces with that property as long as they are perfect, e.g., $\mathbb{Q}\coprod(\mathbb{R}-\mathbb{Q})$.