  We demonstrate that And\'ujar Guerrero, Thomas and Walsberg's results on
definable compactness in o-minimal structures still hold true in definably
complete locally o-minimal structures. As an application, we show that a
definably simple definable topological group which is regular, Hausdorff and
definably compact as a definable topological space is either discrete or
definably connected. We also study the definable quotient of definable
continuous actions by definably compact definable topological groups.
