Uniformly finer topologies

Abstract

We show that both the rationals (a σ-compact metric space) and the irrationals (a polish space) admit a topology, strictly finer than the euclidean at every point, such that the resulting spaces are homeomorphic to the original ones. An example with the same property is also given, where the metric space is complete and locally compact. However, if a topological space is both Čech-complete and σ-compact, then this cannot happen.