Completeness of locally kw-groups and related infinite-dimensional Lie groups

Recall that a topological space X is said to be a k(w)-space if it is the direct limit of an ascending sequence K-1 subset of K-2 subset of . . . of compact Hausdorff topological spaces. If each point in a Hausdorff space X has an open neighbourhood which is a k(w)-space, then X is called locally k(w). We show that a topological group is complete whenever the underlying topological space is locally k(w). As a consequence, every infinite dimensional Lie group modelled on a Silva space is complete. (c) 2017 Elsevier B.V. All rights reserved.