Homeomorphism establishes a very strong relationship between topological spaces. We know that many important properties such as compactness and connectedness are preserved under homeomorphism, and fundamental groups of such spaces are isomorphic.
I wish to gain some more intuition on the possible limits of homeomorphism, so I was wondering if you have any examples of properties that are not preserved under homeomorphism?
Searching SE I found one such example here, which discusses completeness. I think that Vadim's answer was particularly good, since it shows that while the choice of metric is independent of the existence of homeomorphism, the fact that a space $X$ is homeomorphic to a complete metric space means it is metrizable by some complete metric.
So are there any more properties like the above? Or ones that entirely break under homeomorphism?