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

Recall that a topological space X is said to be a kω-space if it is the direct limit of an ascending sequence K1⊆K2⊆⋯ of compact Hausdorff topological spaces. If each point in a Hausdorff space X has an open neighbourhood which is a kω-space, then X is called locally kω. We show that a topological group is complete whenever the underlying topological space is locally kω. As a consequence, every infinite-dimensional Lie group modelled on a Silva space is complete.