On Loeb and sequential spaces in ZF

摘要

A topological space is called Loeb if the collection of all its non-empty closed sets has a choice function. Loeb and sequential spaces are investigated in the absence of the axiom of choice. Among other results, the following theorems are proved in ZF: (i) if X is a Cantor completely metrizable second-countable space, then X-omega is Loeb; (ii) if a sequential, locally sequentially compact space X has the property that every infinite countable collection of non-empty closed subsets of X has a choice function, then the Cartesian product X x Y of X with any sequential space Y is sequential; (iii) if R is sequential, then every second-countable compact Hausdorff space is sequential.Several statements on Loeb and sequential spaces are shown to be independent of ZF. Open problems are posed. (C) 2020 Elsevier B.V. All rights reserved.