Aplicaciones cerradas y espacios en los que componentes y cuasicomponentes coinciden

Abstract

This interesting and well-written survey article is devoted to the class P of topological spaces in which components and quasi-components coincide. This class includes the compact Hausdorff spaces and the locally connected spaces. It also includes every subset of the real line but not every subset of the plane. This class is closed under homotopy type, but the authors state that "it does not seem to be possible to give easily-stated conditions'' for membership in P . They do give some sufficient conditions using the fact that, to any topological space X , one can associate the quotient space ΔX in which each quasi-component is identified to a point (they show that this association is categorically natural). These conditions include the assumption that the quotient map is closed. For example, they show that, if X is normal and ΔX is zero-dimensional, then X∈P . Variations of this include the result that, if ΔX is zero-dimensional and the quasi-components are compact, then X∈P , and the result that, if X is locally compact Lindelöf and Hausdorff, then X∈P . No proofs are given. Can the fact that P is closed under homotopy equivalence be improved by allowing a more arbitrary homotopy index set (or not using a product structure at all)? What is an example of a space X whose quotient map is closed but X∉P ?