Show that 1) $(-1,1)$ is homeomorphic to $R$
2) $(-1,3]$ is homeomorphic to $[1,3)$.
The definition of homeomorphism :- Let $X$ & $Y$ be topological space . A mapping $f$ from $X$ to $Y$ is said to be homeomorphism if f is bijective, continuous and $f^{-1}$ is continuous.
So for $(1)$ I should define a map. Can I define $f$ as $f(x)=x$ ?
And for $(2)$, $f(x )=\frac{(x+1)}{3}+1 ?$
Clearly both functions are bijective and continuous. Also inverse of f exists and continuous.