Transforming Linear Sets into Nullsets

Abstract

Let F be a nowhere dense closed subset of I = [0,1]. It is an almost trivial observation that there exists a homeomorphism h of I onto itself such that h(F) is a nullset. In fact, letting G = I-F, it suffices to take

$$ \rho \left( {f,g} \right) = {\sup_{{a \leqslant x \leqslant b}}}\left| {f(x) - g(x)} \right| $$