A generalization of the Open Mapping Theorem and a possible generalization of the Baire Category Theorem

TL;DR

This paper explores the boundaries of the Open Mapping Theorem by characterizing the continuum as the minimal size of a compact set family that fails the theorem in certain locally compact groups.

Contribution

It introduces a novel characterization of the continuum related to the failure of the Open Mapping Theorem in locally compact groups.

Findings

Continuum characterized as minimal compact cover size

Links between the Open Mapping Theorem and set cardinalities

Potential generalizations of the Baire Category Theorem

Abstract

We characterize continuum as the smallest cardinality of a family of compact sets needed to cover a locally compact group for which the Open Mapping Theorem does not hold.