Question about bijective functions and homeomorphism Is it true that 
"If two metric spaces each of which is the image of the other under a bijective continuous function, then the two metric spaces are homeomorphic."??
Thank you so much!!

I posted a very similar question on mathoverflow (link) and one of the replies there is a reasonably easy metrisable one:
Let $X$ and $Y$ be spaces with underlying set $\mathbb{R}$. On $X$ we put the topology that consists of the usual topology on $(0,\infty)$ and the discrete metric on all $x \le 0$. On $Y$ we put the topology that is the usual one on $(-1,0)$ and $(1,\infty)$ and is discrete everywhere else. 
Now $f(x) = x+1$ is a continuous bijection from $X$ to $Y$ and also a continuous bijection from $Y$ to $X$.
But they are not homeomorphic as $X$ as one non-trivial component, and $Y$ has two.
