homeomorphism from interval $[a,b]$ to $[0,1]\subset \mathbb{R}$

Question

I need to show that every interval $[a,b]$ is homeomorph to $[0,1]\subset \mathbb{R}$. I've found this answer but it only deals with open sets, and I need an answer that deals with closed sets.

Answer 1

Why don't you linearly do it: interval length should be expaded/shrunk, and $a$ must map to $0$, so $$f:[a,b] \longrightarrow [0,1]$$ $$\ \ \ \ \ \ \ \ \ x\longrightarrow \frac{x-a}{b-a}$$ ?

It has a continuous inverse.

Answer 2

Consider $A=\{(x,0) \mid 0 \leq x \leq 1\}$ and $B = \{ (x,1) \mid a \leq x \leq b\}$. Then draw the line thru $(1,0)$ and $(b,1)$. This line intersects the vertical axis somewhere. Use straight lines from this intersection point to construct a homeomorphism of $[0,1]$ onto $[a,b]$.