Closed bounded sets in 1-h-minimal valued fields

TL;DR

This paper demonstrates that 1-h-minimal valued fields exhibit a form of naive compactness for certain definable sets, and applies this to show local definable groups have neighborhoods of the identity that are open subgroups.

Contribution

It establishes a new compactness property in 1-h-minimal fields and applies it to the structure of local definable groups.

Findings

1-h-minimal fields satisfy naive compactness for certain definable sets.

Local topological definable groups have neighborhoods of the identity that are open subgroups.

Abstract

We show that the 1-h-minimal fields satisfy a property of naive compactness for decreasing definable families of closed bounded sets indexed by the value group. We use this to prove that a local topological definable group has a definable family of neighborhoods of the identity consisting of open subgroups.