Isomorphism between [0,1] to (0,1) I need to find an isomorphism between [0,1] to (0,1) can you help me with this please?
thanks. benny

If you just want a bijection there are many ways to do it. Here's a rather contrived one:
Let $\alpha$ be a limit ordinal of the same cardinality as $(0,1)$. By the Cantor-Bernstein-Schröder theorem there exists a bijective map $f: (0,1) \to \alpha$. Let $S$ denote the successor "function" (I know it's not a function - hence the quotes). Note that $\alpha$ does not have a maximal element, so it makes sense to define $\phi: [0,1] \to (0,1)$ by
$$\phi(x) = \begin{cases}
(f^{-1} \circ S \circ S \circ f)(x) & \text{ for $x \in (0,1)$}, \\
f^{-1}(1) & \text{ for $x = 1$}, \\
f^{-1}(0) & \text{ for $x = 0$}.
\end{cases}$$
Showing that $\phi$ is a bijection is an easy exercise. It's also completely useless and nonconstructive. 
