Topology via logic

목차 (Table of Contents)

• CONTENTS
• Preface
• Notation
• 1 Introduction - A Historical Overview = 1
• 2 Affinnative and refutative assertions, In which we see a Logic of Finite Observations and take this as the notion we want to study. = 5
• 3 Frames, In which we set up an algebraic theory for the Logic of Finite Observations: its algebras are frames. = 12
• 3.1 Algebraicizing logic = 12
• 3.2 Posets = 13
• 3.3 Meets and joins = 14
• 3.4 Lattices = 18
• 3.5 Frames = 21
• 3.6 Topological spaces = 22
• 3.7 Some examples from computer science = 23
• Finite observations on bit streams = 23
• Different physical assumptions = 27
• Flat domains = 28
• Function spaces = 29
• 3.8 Bases and subbases = 31
• 3.9 The real line = 32
• 3.10 Complete Heyting algebras = 34
• 4 Frames as algebras, In which we see methods that exploit our algebraicizing of logic. = 38
• 4.1 Semilattices = 38
• 4.2 Generators and relations = 39
• 4.3 The universal characterization of presentations = 42
• 4.4 Generators and relations for frames = 46
• 5 Topology: the definitions, In which we introduce Topological Systems, subsuming topological spaces and locales. = 52
• 5.1 Topological systems = 52
• 5.2 Continuous maps = 54
• 5.3 Topological spaces = 57
• Spatialization = 59
• 5.4 Locales = 60
• Localification = 62
• 5.5 Spatial locales or sober spaces = 64
• 5.6 Sunimary = 67
• 6 New topologies for old, In which we see some ways of constructing topological systems, and some ways of specifying what they construct. = 70
• 6.1 Subsystems = 70
• 6.2 Sublocales = 71
• 6.3 Topological sums = 76
• 6.4 Topological products = 80
• 7 Point logic, In which we seek a logic of points, and find an ordering and a weak disjunction. = 89
• 7.1 The specialization preorder = 89
• 7.2 Directed disjunctions of points = 92
• 7.3 The Scott topology = 95
• 8 Compactness, In which we define conjunctions ofpoints and discover the notion of Compactness. = 98
• 8.1 Scott open filters = 98
• 8.2 The Scott Open Filter Theorem = 100
• 8.3 Compactness and the reals = 103
• 8.4 Examples with bit-streams = 105
• 8.5 Compactness and products = 106
• 8.6 Local compacttiess and function spaces = 110
• 9 Spectral algebraic locales, In which we see a category of locales within which we can do the topology of main theory. = 116
• 9.1 Algebraic posets = 16
• 9.2 Spectral locales = 119
• 9.3 Spectral algebraic locales = 121
• 9.4 Finiteness, second countability and ω-algebraicity = 125
• 9.5 Stone spaces = 127
• 10 Domain Theory, In which we see how certain parts of domain theory can be done topologically. = 134
• 10.1 Why domain theory? = 134
• 10.2 Bottoms and lifting = 136
• 10.3 Products = 138
• 10.4 Sums = 139
• 10.5 Function spaces and Scott domains = 142
• 10.6 Strongly algebraic locales (SFP) = 146
• 10.7 Domain equations = 152
• 11 Power domains, In which we investigate domains of subsets of a given domain. = 165
• 11.1 Non-determinism and sets = 165
• 11.2 The Smyth power domain = 166
• 11.3 Closed sets and the Hoare power domain = 169
• 11.4 'Me Plotkin power domain = 171
• 11.5 Sets implemented as lists = 176
• 12 Spectra of rings, In which we see some old examples of spectral locales. = 181
• 12.1 The Pierce spectrum = 181
• 12.2 Quantales and the Zariski spectrum = 182
• 12.3 Cohn's field spectrum = 185
• Bibliography = 191
• Index = 196