Effros, Baire, Steinhaus and Non-Separability

TL;DR

This paper presents a simplified proof of an enhanced Effros Open Mapping Principle using shift-compactness and Baire property, applicable to a broad class of normed groups, including non-separable ones.

Contribution

It introduces a shift-compactness approach to the Effros Open Mapping Principle, extending its applicability beyond separable spaces and simplifying the proof process.

Findings

Provides a short proof of the Effros Open Mapping Principle

Extends the Open Mapping Theorem to non-separable groups

Introduces a shift-compactness theorem based on Baire property

Abstract

We give a short proof of an improved version of the Effros Open Mapping Principle via a shift-compactness theorem (also with a short proof), involving `sequential analysis' rather than separability, deducing it from the Baire property in a general Baire-space setting (rather than under topological completeness). It is applicable to absolutely-analytic normed groups (which include complete metrizable topological groups), and via a Steinhaus-type Sum-set Theorem (also a consequence of the shift-compactness theorem) includes the classical Open Mapping Theorem (separable or otherwise).