Around definable types in p-adically closed fields

Abstract

We prove some technical results on definable types in p-adically closed fields, with consequences for definable groups and definable topological spaces. First, the code of a definable n-type (in the field sort) can be taken to be a real tuple (in the field sort) rather than an imaginary tuple (in the geometric sorts). Second, any definable type in the real or imaginary sorts is generated by a countable union of chains parameterized by the value group. Third, if X is an interpretable set, then the space of global definable types on X is strictly pro-interpretable, building off work of Cubides Kovacsics, Hils, and Ye [7], [8]. Fourth, global definable types can be lifted (in a non-canonical way) along interpretable surjections. Fifth, if G is a definable group with definable f-generics (dfg), and G acts on a definable set X, then the quotient space $X/G$ is definable, not just interpretable. This explains some phenomena observed by Pillay and Yao [24]. Lastly, we show that interpretable topological spaces satisfy analogues of first-countability and curve selection. Using this, we show that all reasonable notions of definable compactness agree on interpretable topological spaces, and that definable compactness is definable in families.