Topological Fields and Near Valuations

PART I: SURVEY OF TOPOLOGICAL FIELDS 1 INTRODUCTION 1.1 Neighborhood Bases at Zero 1.2 Alternate Axiomatizations 1.3 Basic Properties 2 VALUATIONS AND OTHER EXAMPLES 2.1 Prevaluations 2.2 Examples of Valued Fields 3 THE LATTICE OF RING TOPOLOGIES 3.1 Lattices of Topologies . . . 3.2 Weakening Ring Topologies 3.3 Minimal Topologies 3.4 Independence 4 LOCALLY BOUNDED FIELDS 4.1 Bounded Sets 4.2 Locally Bounded Rings 4.3 Preorders 4.4 Preorders and Topologies 4.5 Lattice Results 5 NORMED FIELDS 5.1 Norms 5.2 Nilpotence and Normability 6 COMPLETENESS 6.1 Completions of Rings 6.2 Completions of Fields 7 EMBEDDING AND EXTENSION 7.1 The Problem 7.2 The Product Topology Extension 8 EXISTENCE OF FIELD TOPOLOGIES 9 CONNECTED FIELDS 10 DISCONNECTED FIELDS 10.1 Extremally Disconnected Fields 10.2 Ultraregular Fields . 11 LINEAR FIELDS PART 11: VALUED FIELDS 12 ABSOLUTE VALUES 12.1 Nonarchimedean Absolute Values 12.2 Absolute Values on PID's 12.3 Equivalent Absolute Values 12.4 Equivalent Valuations 12.5 Powers of Absolute Values 13 PLACES 14 VECTOR SPACES AND STRICTLY MINIMAL FIELDS .14.1 Strictly Minimal Fields .14.2 Completeness and Norms 14.3 Normed Algebras 15 EXTENSIONS OF VALUATIONS 15.1 Existence of Extensions 15.2 Archimedean Absolute Values 15.3 Complete and Algebraically Closed Fields . 16 CHARACTERIZATIONS 16.1 Topologies Induced by Absolute Values 16.2 Archimedean Valuations 16.3 Type V Fields 16.4 Addiator Sequences 16.5 Topologies Induced by Valuations