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# Prove that two closed intervals are homeomorphic

From Mathematics Is A Science

Jump to navigationJump to searchQuestion: Prove that there is a homeomorphism between [a,b] and [c,d], and between (a,b) and (c,d).

Solution: The function $f:[a,b]\rightarrow [c,d]$ can be linear with $f(a)=c,f(b)=d$. The slope is $m=\frac{d-c}{b-a}$ . Now use the point-slope form: $f(x)=c+m(x-a)$. It's continuous and so is its inverse (what's the formula?). For the open intervals, just take the restriction of this function.

- What about a homeomorphism between [a,b] and (c,d)?
- What about a homeomorphism between [a,b] and [c,d)?