Construct an explicit bijection g: [0,1] --> (0,1).
Hint: One way to do it would be three parts. One for g(0), one for g(1/n) for n belongs to N and one for everything else.
A pretty standard example is this: the set of all rational numbers in (0, 1) is countable so can be "listed" $\displaystyle \{r_1, r_2, ..., r_n, ...\}$. Map 0 to $\displaystyle r_1$, 1 to $\displaystyle r_2$, and for all rational numbers $\displaystyle a_i$ to $\displaystyle a_{i+2}$. All irrational numbers in (0, 1) are mapped to themselves.