-1
$\begingroup$

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?

$\endgroup$
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.