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## Ehud Hrushovski and François Loeser

Print publication date: 2016

Print ISBN-13: 9780691161686

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691161686.001.0001

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# Definable compactness

Chapter:
(p.57) Chapter Four Definable compactness
Source:
Non-Archimedean Tame Topology and Stably Dominated Types (AM-192)
Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691161686.003.0004

This chapter describes the notion of definable compactness for subsets of unit vector V. One of the main results is Theorem 4.2.20, which establishes the equivalence between being definably compact and being closed and bounded. The chapter gives a general definition of definable compactness that may be useful when the definable topology has enough definable types. The o-minimal formulation regarding limits of curves is replaced by limits of definable types. The chapter relates definable compactness to being closed and bounded and shows that the expected properties hold. In particular, the image of a definably compact set under a continuous definable map is definably compact.

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