Non-Archimedean Tame Topology and Stably Dominated Types (AM-192)

Contents\n1 Introduction\n2 Preliminaries\n	2.1 Definable sets\n	2.2 Pro-definable and ind-definable sets\n	2.3 Definable types\n	2.4 Stable embeddedness\n	2.5 Orthogonality to a definable set\n	2.6 Stable domination\n	2.7 Review of ACVF\n	2.8 Г-internal sets\n	2.9 Orthogonality to Г\n	2.10 V̂ for stable definable V\n	2.11 Decomposition of definable types\n	2.12 Pseudo-Galois coverings\n3 The space V̂ of stably dominated types\n	3.1 V̂ as a pro-definable set\n	3.2 Some examples\n	3.3 The notion of a definable topological space\n	3.4 V̂ as a topological space\n	3.5 The affine case\n	3.6 Simple points\n	3.7 v-open and g-open subsets, v+g-continuity\n	3.8 Canonical extensions\n	3.9 Paths and homotopies\n	3.10 Good metrics\n	3.11 Zariski topology\n	3.12 Schematic distance\n4 Definable compactness\n	4.1 Definition of definable compactness\n	4.2 Characterization of definable compactness\n5 A closer look at the stable completion\n	5.1 A^n and spaces of semi-lattices\n	5.2 A representation of ℙ^n\n	5.3 Relative compactness\n6 Г-internal spaces\n	6.1 Preliminary remarks\n	6.2 Topological structure of Г-internal subsets\n	6.3 Guessing definable maps by regular algebraic maps\n	6.4 Relatively Г-internal subsets\n7 Curves\n	7.1 Definability of Ĉ for a curve C\n	7.2 Definable types on curves\n	7.3 Lifting paths\n	7.4 Branching points\n	7.5 Construction of a deformation retraction\n8 Strongly stably dominated points\n	8.1 Strongly stably dominated points\n	8.2 A Bertini theorem\n	8.3 Г-internal sets and strongly stably dominated points\n	8.4 Topological properties of V^#\n9 Specializations and ACV^2F\n	9.1 g-topology and specialization\n	9.2 v-topology and specialization\n	9.3 ACV^2F\n	9.4 The map R^20 21 : V̂20 → V̂21\n	9.5 Relative versions\n	9.6 g-continuity criterion\n	9.7 Some applications of the continuity criteria\n	9.8 The v-criterion on V̂\n	9.9 Definability of v- and g-criteria\n10 Continuity of homotopies\n	10.1 Preliminaries\n	10.2 Continuity on relative ℙ^1\n	10.3 The inflation homotopy\n	10.4 Connectedness and the Zariski topology\n11 The main theorem\n	11.1 Statement\n	11.2 Proof of Theorem 11.1.1: Preparation\n	11.3 Construction of a relative curve homotopy\n	11.4 The base homotopy\n	11.5 The tropical homotopy\n	11.6 End of the proof\n	11.7 Variation in families\n12 The smooth case\n	12.1 Statement\n	12.2 Proof and remarks\n13 An equivalence of categories\n	13.1 Statement of the equivalence of categories\n	13.2 Proof of the equivalence of categories\n	13.3 Remarks on homotopies over imaginary base sets\n14 Applications to the topology of Berkovich spaces\n	14.1 Berkovich spaces\n	14.2 Retractions to skeleta\n	14.3 Finitely many homotopy types\n	14.4 More tame topological properties\n	14.5 The lattice completion\n	14.6 Berkovich points as Galois orbits\nBibliography\nIndex\nList of notations