Let $X$ and $Y$ be complete metric spaces and let $T:X\to Y$ bijective and continuous. Will $T$ be a homeomorphism?

Question

Let $X$ and $Y$ be complete metric spaces and let $T:X\to Y$ bijective and continuous. Will $T$ be a homeomorphism?

Can anyone give me a hint?

Answer 1

Hint:

Have a look at the identity function on $\mathbb R$ where the domain is equipped with discrete topology and the codomain with usual topology.

Answer 2

In general, no. Take $f\colon\mathbb{Z}_+\longrightarrow\{0\}\cup\left\{\frac1n\,|\,n\in\mathbb N\right\}$ be the function such that $f(0)=0$ and that $f(n)=\frac1n$ if $n\in\mathbb N$. Endow both $\mathbb{Z}_+$ and $\{0\}\cup\left\{\frac1n\,|\,n\in\mathbb N\right\}$ with the usual topology. Then $f$ is continuous and bijective, but $f^{-1}$ is discontinuous at $0$.