I need to show that any open interval is homeomorphic to the real line.
I know that $f(x)=a+e^x$ will work for the mapping $f:R \to (a,\infty)$ and $f(x)=b-e^{-x}$ will work for the mapping $f:R \to (-\infty,b).$
Without using two functions, how can I prove the result in general?
Best Answer
Let $$ f(x) = \frac{x}{x^2-1}. $$ This is a homeomorphism from $(-1,1)$ to $\mathbb R$.
Let $$ g(x) = \frac{1}{1+2^{-x}}. $$ That is a homeopmorphism from $\mathbb R$ to $(0,1)$.
For any other bounded interval $(a,b)$, just rescale and relocate.