Completeness of locally $k_\omega$-groups and related infinite-dimensional Lie groups

TL;DR

This paper proves that topological groups with locally $k_ extomega$ spaces are complete, and as a result, infinite-dimensional Lie groups modeled on Silva spaces are also complete, advancing understanding of their topological structure.

Contribution

It establishes the completeness of topological groups with locally $k_ extomega$ spaces and applies this to infinite-dimensional Lie groups modeled on Silva spaces.

Findings

Topological groups with locally $k_ extomega$ spaces are complete.

Infinite-dimensional Lie groups modeled on Silva spaces are complete.

Abstract

Recall that a topological space is said to be a $k_\omega$-space if it is the direct limit of an ascending sequence of compact Hausdorff topological spaces. If each point in a Hausdorff space $X$ has an open neighbourhood which is a $k_\omega$-space, then $X$ is called locally $k_\omega$. We show that a topological group is complete whenever the underlying topological space is locally $k_\omega$. As a consequence, every infinite-dimensional Lie group modelled on a Silva space is complete.