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?
Best Answer
What about a sequence being a Cauchy sequence? The spaces $\mathbb R$ and $(0,+\infty)$ are homeomorphic (with respect to the usual metric). A homeomorphism between them is $\exp\colon\mathbb{R}\longrightarrow(0,+\infty)$, whose inverse is $\log$. The sequence $(1/n)_{n\in\mathbb N}$ is a Cauchy sequence in $(0,+\infty)$, but $\bigl(\log(1/n)\bigr)_{n\in\mathbb N}$ is not.