I know that two metric spaces are homeomorphic if there is a function from one to another such that $f$ is continuous, one-to-one, onto, with continuous inverse $f^{-1}$.
I know how to prove two metric spaces are homeomorphic.
I don't understand how to prove that two metric spaces are not homeomorphic.
I have to prove using knowledge I have of open sets, closed sets, limit and continuity.
I cannot use compactness, disconnectedness, etc.
Best Answer
Let $f\colon \mathbb{R}\to\mathbb{R}_d$ be a homeomorphism, where the domain has the usual metric and the codomain the discrete metric.
The sequence $(1/n)_{n>0}$ converges to $0$ in $\mathbb{R}$ with the usual metric, so $(f(1/n))_{n>0}$ should converge to $f(0)$ in $\mathbb{R}_d$. But in a discrete space only the eventually constant sequences converge.