Inductive limits of quasi locally Baire spaces

TL;DR

This paper investigates properties of inductive limits of quasi-locally Baire spaces, extending existing theorems on regularity and providing new conditions under which these limits retain desirable topological features.

Contribution

It generalizes the concept of quasi-locally Baire spaces and extends Qiu's theorem on regularity of inductive limits of strictly webbed spaces.

Findings

Inductive limit of strictly webbed spaces is regular if quasi-locally Baire.

If each step is strictly webbed and quasi-locally Baire, the inductive limit is quasi-locally Baire when regular.

Provides examples illustrating the theoretical results.

Abstract

We study quasi-locally complete locally convex spaces and generalize this concept to quasi-locally Baire locally convex spaces. It is shown that an inductive limit of strictly webbed spaces is regular if it is quasi-locally Baire. This extends Qiu's theorem on regularity. Additionally, if each step is strictly webbed and quasi-locally Baire, then the inductive limit is quasi-locally Baire if it is regular. Relevant examples are provided.