Question: show that the metric space $(\mathbb{R^n}, d)$ is connected. Where $d$ is maximum metric (chebyshev distance)
Given Hint: every non-empty proper subset of $\mathbb{R^n}$ has a non-empty boundary.
My attempt: I know $\mathbb{R^n}$ with a usual distance metric is connected. Further I know that, these two metrics (usual metric and maximum metric) on $\mathbb{R^n}$ are equivalent and hence $\mathbb{R^n}$ with respect to maximum metric is connected.
I don't know my attempt is correct or not. Further I like two know if the two metrics are equivalent then what can we say?
I didn't used the hint! How can i use it?
Please help.
Best Answer
Indeed you have the right idea.
$(\Bbb{R}^k ,d_2)$ is homeomorphic to $(\Bbb{R}^k ,d_{\infty})$ since the two metrics are equivalent.
You do not need to use the hint at all.