2
$\begingroup$

It is well known that compact Hausdorff topologies on a set are all incomparable; i.e. if $X$ is a set and $\tau_1$ and $\tau_2$ are compact Hausdorff topologies on $X$ then the identity map $(X,\tau_1)\to (X,\tau_2)$ is continuous if and only if $\tau_1 =\tau_2$. Is the same true for the $\sigma$-compact Hausdorff case? What if the spaces are also locally compact?

$\endgroup$
  • 1
    $\begingroup$ Embed $\mathbb{N}\setminus$ into $\mathbb{R}$ via a) the identity, b) $n \mapsto 1/n$ for $n > 0$ and $0 \mapsto 0$. The topologies are comparable, both are locally compact $\sigma$-compact Hausdorff topologies, but not identical. $\endgroup$ – Daniel Fischer Nov 2 '17 at 11:14
0
$\begingroup$

You can use $\mathbb{Z}$ with the discrete and the finite complement topologies.

They are both $\sigma$-compact and Haussdorff, but not the same.

$\endgroup$
  • 2
    $\begingroup$ The finite complement topology is not Hausdorff. $\endgroup$ – Daniel Fischer Nov 2 '17 at 11:09

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.