A non-archimedean Dugundji extension theorem

Ka̧kol, Jerzy; Kubzdela, Albert; Śliwa, Wiesław

doi:10.1007/s10587-013-0010-8

Cited by 7 publications

(4 citation statements)

References 15 publications

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“…The earlier techniques and approaches to the purely topological versions of the above problems cannot be carried over to the definable settings because, among others, non-Archimedean geometry over non-locally compact fields suffers from lack of definable Skolem functions. Perhaps the strongest, purely topological, non-Archimedean results on retractions are those from the papers [11,17] recalled below.…”

Section: ] and [4 Section 52])mentioning

confidence: 99%

A closedness theorem over Henselian fields with analytic structure and its applications

Nowak

2020

Banach Center Publ.

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In this brief note, we present our closedness theorem in geometry over Henselian valued fields with analytic structure. It enables, among others, application of resolution of singularities and of transformation to normal crossings by blowing up in much the same way as over locally compact ground fields. Also given are many applications which, at the same time, provide useful tools in geometry and topology of definable sets and functions. They include several versions of the Łojasiewicz inequality, Hölder continuity of definable functions continuous on closed bounded subsets of the affine space, piecewise continuity of definable functions or curve selection. We also present our most recent research concerning definable retractions and the extension of continuous definable functions. These results were established in several successive papers of ours, and their proofs made, in particular, use of the following fundamental tools: elimination of valued field quantifiers, term structure of definable functions and b-minimal cell decomposition, due to Cluckers-Lipshitz-Robinson, relative quantifier elimination for ordered abelian groups, due to Cluckers-Halupczok, the closedness theorem as well as canonical resolution of singularities and transformation to normal crossings by blowing up due to Bierstone-Milman. As for the last tool, our approach requires its definable version established in our most recent paper within a category of definable, strong analytic manifolds and maps.

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Section: ] and [4 Section 52])mentioning

confidence: 99%

A closedness theorem over Henselian fields with analytic structure and its applications

Nowak

2020

Banach Center Publ.

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“…Perhaps the strongest, purely topological, non-Archimedean results on retractions are those from the papers [22,28] recalled below respectively. Theorem 4.…”

Section: Remarkmentioning

confidence: 99%

“…Furthermore, the earlier techniques and approaches to the purely topological versions of those problems cannot be carried over to the definable settings because, among others, non-Archimedean geometry over non-locally compact fields suffers from lack of definable Skolem functions. For a more detailed discussion about their classical, purely topological counterparts (see e.g., [20][21][22]), we refer the reader to our paper [18].…”

Section: Introductionmentioning

confidence: 99%

Definable Transformation to Normal Crossings over Henselian Fields with Separated Analytic Structure

Nowak

2019

Symmetry

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We are concerned with rigid analytic geometry in the general setting of Henselian fields K with separated analytic structure, whose theory was developed by Cluckers–Lipshitz–Robinson. It unifies earlier work and approaches of numerous mathematicians. Separated analytic structures admit reasonable relative quantifier elimination in a suitable analytic language. However, the rings of global analytic functions with two kinds of variables seem not to have good algebraic properties such as Noetherianity or excellence. Therefore, the usual global resolution of singularities from rigid analytic geometry is no longer at our disposal. Our main purpose is to give a definable version of the canonical desingularization algorithm (the hypersurface case) due to Bierstone–Milman so that both of these powerful tools are available in the realm of non-Archimedean analytic geometry at the same time. It will be carried out within a category of definable, strong analytic manifolds and maps, which is more flexible than that of affinoid varieties and maps. Strong analytic objects are those definable ones that remain analytic over all fields elementarily equivalent to K. This condition may be regarded as a kind of symmetry imposed on ordinary analytic objects. The strong analytic category makes it possible to apply a model-theoretic compactness argument in the absence of the ordinary topological compactness. On the other hand, our closedness theorem enables application of resolution of singularities to topological problems involving the topology induced by valuation. Eventually, these three results will be applied to such issues as the existence of definable retractions or extending continuous definable functions. The established results remain valid for strictly convergent analytic structures, whose classical examples are complete, rank one valued fields with the Tate algebras of strictly convergent power series. The earlier techniques and approaches to the purely topological versions of those issues cannot be carried over to the definable settings because, among others, non-Archimedean geometry over non-locally compact fields suffers from lack of definable Skolem functions.

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“…(5) Every locally compact totally disconnected orderable space is retractifiable. Kąkol-Kubzdela-Śliwa [33] showed that every compact metrizable subspace Y of an ultraregular space X is a retract of X.…”

Section: Introductionmentioning

confidence: 99%

Simultaneous extensions of metrics and ultrametrics of high power

Ishiki¹

2022

Preprint

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In this paper, generalized metrics mean metrics taking values in general linearly ordered Abelian groups. Using the Hahn fields, we first prove that for every generalized metric space, if the set of the Archimedean equivalence classes of the range group of the metric has an infinite decreasing sequence, then every nonempty closed subset of the space is a uniform retract of the ambient space. Next we construct simultaneous extensions of generalized metrics and ultrametrics. From the existence of extensors of generalized metrics, we characterize the final compactness of generalized metrizable spaces using the completeness of generalized metrics. Contents 1. Introduction 1 2. Preliminaries 11 3. Retractions 28 4. Extensors of ultrametrics and metrics of high power 32 5. Table of symbols 39 References 41

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A non-archimedean Dugundji extension theorem

Cited by 7 publications

References 15 publications

A closedness theorem over Henselian fields with analytic structure and its applications

A closedness theorem over Henselian fields with analytic structure and its applications

Definable Transformation to Normal Crossings over Henselian Fields with Separated Analytic Structure

Simultaneous extensions of metrics and ultrametrics of high power

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