Abstract

We characterize the notion of definable compactness for topological spaces definable in o-minimal structures, answering questions posed by Peterzil and Steinhorn (J. London Math. Soc. $(2)$ 59:3 (1999), 769–786) and Johnson (J. Symb. Log. 83:4 (2018), 1477–1500). Specifically, we prove the equivalence of various definitions of definable compactness in the literature, including those in terms of definable curves, definable types and definable downward directed families of closed sets.

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