A closedness theorem and applications in geometry of rational points over Henselian valued fields

We develop geometry of algebraic subvarieties of $K^{n}$ over arbitrary Henselian valued fields $K$. This is a continuation of our previous article concerned with algebraic geometry over rank one valued fields. At the center of our approach is again the closedness theorem that the projections $K^{n} \times \mathbb{P}^{m}(K) \to K^{n}$ are definably closed maps. It enables application of resolution of singularities in much the same way as over locally compact ground fields. As before, the proof of that theorem uses i.a. the local behavior of definable functions of one variable and fiber shrinking, being a relaxed version of curve selection. But now, to achieve the former result, we first examine functions given by algebraic power series. All our previous results will be established here in the general settings: several versions of curve selection (via resolution of singularities) and of the Łojasiewicz inequality (via two instances of quantifier elimination indicated below), extending continuous hereditarily rational functions as well as the theory of regulous functions, sets and sheaves, including Nullstellensatz and Cartan's theorems A and B. Two basic tools applied in this paper are quantifier elimination for Henselian valued fields due to Pas and relative quantifier elimination for ordered abelian groups (in a many-sorted language with imaginary auxiliary sorts) due to Cluckers--Halupczok. Other, new applications of the closedness theorem are piecewise continuity of definable functions, Holder continuity of definable functions on closed bounded subsets of $K^{n}$, the existence of definable retractions onto closed definable subsets of $K^{n}$, and a definable, non-Archimedean version of the Tietze--Urysohn extension theorem. In a recent preprint, we established a version of the closedness theorem over Henselian valued fields with analytic structure along with some applications.