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The paper deals with Henselian valued field with analytic structure.Actually, we are focused on separated analytic structures, but the resultsremain valid for strictly convergent analytic ones as well. A classical exampleof the latter is a complete rank one valued fields with the Tate algebra ofstrictly convergent power series. The algebraic case was treated in ourprevious papers. Here we are going to carry over the research to the generalanalytic settings. Also considered are the local rings of analytic functiongerms induced by a given separated Weierstrass system, which turn out to beexcellent and regular. Several results are established as, for instance,piecewise continuity of definable functions, several versions of the\L{}ojasiewicz inequality, H\"{o}lder continuity of definable functionscontinuous on closed bounded subsets of the affine space and curve selectionfor definable sets. Likewise as before, at the center of our approach is theclosedness theorem to the effect that every projection with closed boundedfiber is a definably closed map. It enables application of resolution ofsingularities and of transformation to a normal crossing by blowing up (hereapplied to the local rings of analytic function germs) in much the same way asover locally compact ground fields. We rely on elimination of valued fieldquantifiers, term structure of definable functions and b-minimal celldecomposition, due to Cluckers--Lipshitz--Robinson, as well as on relativequantifier elimination for ordered abelian groups, due to Cluckers--Halupczok.Besides, other two ingredients of the proof of the closedness theorem areexistence of the limit (after finite partitioning of the domain) for adefinable function of one variable and fiber shrinking, being a relaxed versionof curve selection.

Some results of geometry over Henselian fields with analytic structure