Does the following bijection work:
Take any point $(x,y) \in (0,1) \times (0,1).$ Each real number $r \in (0,1)$ may be represented by an infinitely-long decimal expansion (0.235, for example, is the same as 0.234999999...). Take the real numbers $x,y \in \mathbb{R}$ and interlace their decimal expansions to produce a unique real number $r' \in (0,1).$ The number $r'$ being unique, the mapping is 1-1. Given a real number $r' \in (0,1),$ one can unlace the decimal expansion of that number according to the pattern set by the mapping and produce the real numbers $x$ and $y$, and therefore arrive at a unique point $(x,y) \in (0,1) \times (0,1).$
Does a bijection exist?