Existence of Certain Analytic Homeomorphisms

Extract

This note has its origin in the following problem: do there exist non-trivial increasing continuous functions on [0, l] to [0, l], which map the following sets in [0, l] onto themselves: the rational, the algebraic and the transcendental numbers? One such function is obviously f(x) = x; more generally, f(x) = (c + l)x/(cx + 1), with c rational and non-negative, satisfies the conditions. Let G denote the space of order-preserving homeomorphisms of [0, l} onto [0, l], in the uniform metric. It follows from Theorem 1 below that the set S of all such functions i s dense in G. S is clearly a subgroup of G and one may ask what a r e its group-theoretic properties. We shall not consider these questions.