I've been wondering about "topologically equivalent" for some time now. For example:
$S^1$ is "topologically equivalent" to $\mathbb{R}P^1$.
I see that they are homotopy equivalent. But are they also homeomorphic? Probably yes.
Is there a failsafe way to determine whether in a given case "topologically equivalent" means "homeomorphic" or "homotopy equivalent"? Thanks for your help!