Compact Hausdorff Topologies on a set are all incomparable. Is the same true for the $\sigma$-compact Hausdorff case? 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? 

You can use $\mathbb{Z}$ with the discrete and the finite complement topologies.
They are both $\sigma$-compact and Haussdorff, but not the same. 
