Homeomorphism between $[0,1]$ and the extended real line $\mathbb{R} \cup \{\pm \infty\}$

Question

I know that $[-1,1]$ and the extended real line $\overline{\mathbb{R}}$ are homeomorphic. Is it also true that $[0,1]$ and $\overline{\mathbb{R}}$ are homeomorphic? If so how can we show that?

Answer 1

You can find a homeomorphism between $[0,1]$ and $[-1,1]$. Since compositions of bijective resp. continuous functions are bijective resp. continuous, this will imply that $[0,1]$ and $\overline{\mathbb{R}}$ are homeomorphic.

Answer 2

Let $f:[0,1]\rightarrow[-\infty,\infty]$ be a function defined by $$ f(x)=\begin{cases} \tan(\pi(x-\frac{1}{2})), & \mbox{ if }x\in(0,1)\\ \infty, & \mbox{ if }x=1\\ -\infty, & \mbox{ if }x=0 \end{cases}. $$ It can be verified that $f$ is a homeomorphism.

Answer 3

Define the function $f(x) = \tan ( \pi /2) x$ for $0<x<1$ and $f(-1)= -\infty $ , $f(1)= \infty$

The function f is a homeomorphism from $[-1,1]$ to $ \overline{\mathbb{R}}$

We know that $[0,1]$ is homeomophic to $ [-1,1]$