A vast amount of simplification results if one instead works with the Cantor space formula_10, and functions formula_72 Some caution is advised, as the map formula_35 is defined on the unit interval of the real number line, assuming the natural topology on the reals. By contrast, the map formula_36 is defined on the Cantor space formula_10, which, by convention is given a very different topology, the product topology. There is a potential clash of topologies; some care must be taken. However, as presented above, there is a homorphism from the Cantor set into the reals; fortunately, it maps open sets into open sets, and thus preserves notions of continuity.