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?

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    $\begingroup$ 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

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

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

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

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