Topology Via Logic

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Cambridge University Press, Aug 22, 1996 - Computers - 200 pages
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This advanced textbook on topology has three unusual features. First, the introduction is from the locale viewpoint, motivated by the logic of finite observations: this provides a more direct approach than the traditional one based on abstracting properties of open sets in the real line. Second, the author freely exploits the methods of locale theory. Third, there is substantial discussion of some computer science applications. As computer scientists become more aware of the mathematical foundations of their discipline, it is appropriate that such topics are presented in a form of direct relevance and applicability. This book goes some way towards bridging the gap for computer scientists.
 

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Contents

Introduction A Historical Overview
1
Affirmative and refutative assertions In which we see a Logic of Finite Observations and take this as the notion we want to study
5
Frames In which we set up an algebraic theory for the Logic of Finite Observations its algebras are frames
12
32 Posets
13
33 Meets and joins
14
34 Lattices
18
35 Frames
21
36 Topological spaces
22
72 Directed disjunctions of points
92
73 The Scott topology
95
Compactness In which we define conjunctions of points and discover the notion of compactness
98
82 The HofmannMislove Theorem
100
83 Compactness and the reals
104
84 Examples with bit streams
105
85 Compactness and products
106
86 Local compactness and function spaces
110

37 Some examples from computer science
23
Different physical assumptions
27
Flat domains
28
Function spaces
29
38 Bases and subbases
31
39 The real line
32
310 Complete Heyting algebras
34
Frames as algebras In which we see methods that exploit our algebraicizing of logic
38
42 Generators and relations
39
43 The universal characterization of presentations
42
44 Generators and relations for frames
46
Topology the definitions In which we introduce Topological Systems subsuming topological spaces and locales
52
52 Continuous maps
54
53 Topological spaces
57
Spatialization
59
54 Locales
60
Localification
62
55 Spatial locales or sober spaces
65
56 Summary
68
New topologies for old In which we see some ways of constructing topological systems and some ways of specifying what they construct
70
62 Sublocales
71
63 Topological sums
76
64 Topological products
80
Point logic In which we seek a logic of points and find an ordering and a weak disjunction
89
Spectral algebraic locales In which we see a category of locales within which we can do the topology of domain theory
116
92 Spectral locales
119
93 Spectral algebraic locales
121
94 Finiteness second countability and coalgebraicity
125
95 Stone spaces
128
Domain Theory In which we see how certain parts of domain theory can be done topologically
134
102 Bottoms and lifting
136
103 Products
138
104 Sums
139
105 Function spaces and Scott domains
142
106 Strongly algebraic locales SFP
147
107 Domain equations
152
Power domains In which we investigate domains of subsets of a given domain
165
112 The Smyth power domain
166
113 Closed sets and the Hoare power domain
169
114 The Plotkin power domain
171
115 Sets implemented as lists
176
Spectra of rings In which we see some old examples of spectral locales
181
122 Quantales and the Zariski spectrum
182
123 Cohns field spectrum
185
Bibliography
191
Index
196
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Roy L. Crole
Limited preview - 1993
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