Powers of countable Fréchet spaces

Abstract

We examine problems of Galvin and Nogura about preservation of the Fréchet property when taking finite or countably infinite powers of countable topological spaces and groups. It is well known that adding the requirement that the topologies of the given countable spaces are analytic avoids many of the pathologies in this area. Here we show that a set-theoretic principle about open graphs could serve a similar purpose. For example, we show using this principle that if for some n ≥ 2 the power X n of a countable space is Fréchet then so is X n+1 provided it is sequential. We also give an example showing that in some sense this result is optimal.