Show that two spaces are homeomorphic.

Question

Let $X=[0,+ \infty[ $ and $d(x,y)=|\frac{1}{1+x^2}- \frac{1}{1+y^2}| $

1) Show that $(X,d)$ and $(]0,1], d_{2})$ are homeomorphic (where $d_{2}=|x-y|$)

2) Is the space $(X,d)$ connected? compact? complete ?

To do this it means we need to find any function such as:

$f\colon (\mathbb{R}^+, d)\to (]0,1], d_{2})$ is bijective continuous and whose bijection is also continuous.

Does it mean I can take $f(x)=\sin(x) $ which is bijective continuous and of continuous bijection? Is there a way to find the "neatest" homeomorphism?

Answer 1

My guess is that you meant $[0,+\infty)$ instead of $[0,+\infty]$. If that is so, then just define $f(x)=\frac1{1+x^2}$. Then, it is obvious that$$\bigl|f(x)-f(y)\bigr|=d(x,y).$$So, $(X,d)$ is connected, but it is neither compact nor complete.