Tame topology in Hensel minimal structures

Abstract

We are concerned with topology of Hensel minimal structures on non-trivially valued fields K, whose axiomatic theory was introduced in a recent paper by Cluckers–Halupczok–Rideau. We additionally require that every definable subset in the imaginary sort RV, binding together the residue field Kv and value group vK, be already definable in the plain valued field language. This condition is satisfied by several classical tame structures on Henselian fields, including Henselian fields with analytic structure, V-minimal fields, and polynomially bounded o-minimal structures with a convex subring. In this article, we establish many results concerning definable functions and sets. These are, among others, existence of the limit for definable functions of one variable, a closedness theorem, several non-Archimedean versions of the Łojasiewicz inequalities, an embedding theorem for regular definable spaces, and the definable ultranormality and ultraparacompactness of definable Hausdorff LC-spaces.