1
$\begingroup$

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

?

$\endgroup$
8
$\begingroup$

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

$\endgroup$
  • 1
    $\begingroup$ Don't you need infinitely many half-open and infinitely many open intervals? $\endgroup$ – Matthias Wendt Nov 9 '17 at 11:50
5
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.