Real-valued measurable cardinals and sequentially continuous homomorphisms

TL;DR

This paper explores the relationship between measurable cardinals and topological group properties, introducing the concept of g-sequential cardinals and analyzing their implications for group structures and homomorphisms.

Contribution

It introduces the notion of g-sequential cardinals, characterizes g-sequential groups, and links set-theoretic assumptions with topological group properties and homomorphism continuity.

Findings

Existence of real-valued measurable cardinals implies certain topological group class distinctions.

Characterization of g-sequential groups via local weight and cardinality.

Every compact group with Ulam-measurable cardinality admits a finer countably compact topology.

Abstract

A.V.Arkhangel'skii asked in 1981 if the variety $\mathfrak V$ of topological groups generated by free topological groups on metrizable spaces coincides with the class of all topological groups. We show that if there exists a real-valued measurable cardinal then the variety $\mathfrak V$ is a proper subclass of the class of all topological groups. A topological group $G$ is called $g$-sequential if for any topological group $H$ any sequentially continuous homomorphism $G\to H$ is continuous. We introduce the concept of a $g$-sequential cardinal and prove that a locally compact group is $g$-sequential if and only if its local weight is not a $g$-sequential cardinal. The product of a family of non-trivial $g$-sequential topological groups is $g$-sequential if and only if the cardinal of this family is not $g$-sequential. Suppose $G$ is either the unitary group of a Hilbert space or the group of all self-homeomorphisms of a Tikhonov cube. Then $G$ is $g$-sequential if and only if its weight is not a $g$-sequential cardinal. Every compact group of Ulam-measurable cardinality admits a strictly finer countably compact group topology.