Continuous surjections between non-homeomorphic spaces It is very easy to give examples of topological spaces $A$ and $B$ that are not homeomorphic to each other, with a continuous injection from $A$ to $B$, and a continuous injection from $B$ to $A$. See, for example $(0,1)$ and $[0,1]$, as demonstrated in the answer to a similar question on this forum, or this related question. In other words, there exist topological spaces that do not satisfy the Cantor-Schroeder-Bernstein property, when considering injective maps. I was wondering if this fails when considering surjective maps as well. That is, given two topological spaces $A$ and $B$, if there exists a continuous surjection from $A$ to $B$, and a continuous surjection from $B$ to $A$, must $A$ and $B$ be homeomorphic to each other? I think this shouldn't be true: there should be a simple counterexample, but I just can't find one! It would be great to have a counterexample for locally compact Hausdorff spaces, because I want to eventually understand problems with the Cantor-Schroeder-Bernstein property for $C^\star$ algebras...

There is a simple example with $A$ and $B$  both compact!
$[0,2\pi]$ is not homeomorphic to $S^{1}$. The function $t \mapsto (\cos t,\sin t)$ is a  continuous surjection from $[0,2\pi]$ onto $S^{1}$, while the function $(x,y) \mapsto \pi (1+x)$ is a  continuous surjection from $S^{1}$ onto $[0,2\pi]$.
