Definable compactness in definably complete locally o-minimal structures

We show that Andújar Guerrero, Thomas and Walsberg’s results on definable compactness in o-minimal structures still hold in definably complete locally o-minimal structures. As an application, we show that a definably simple definable metric group which is 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.