  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.
