Step functions from the half-open unit interval into a topological space

Abstract

The set of continuous-from-the-right step functions from the half-open unit interval[0, 1[into a topological space X is denoted by X ∗. Elsewhere a topology has been defined which makes X ∗ a contractible, locally contractible space with the subspace of constant functions being homeomorphic to X. When X has a bounded metric ϱ, the topology of X ∗ may be described by the metric d>(f,g)= ∫ 0 1 ρ(f(t),g(t))dt . It is shown here that if X is separable, then X ∗ is separable and if X satisfies the first (or second) axiom of countability, then X ∗ satisfies it too. In contrast, it is shown that properties such as normality do not extend from X to X ∗. This follows from the main result: X ∗ is homeomorphic to its square, and thus contains a copy of X× X (which is closed when X is Hausdorff). The final theorem states that if X has at least two points then X ∗ is not complete metrizable.