# Topology Via Logic

Cambridge University Press, Aug 22, 1996 - Computers - 200 pages
Now in paperback, Topology via Logic is an advanced textbook on topology for computer scientists. Based on a course given by the author to postgraduate students of computer science at Imperial College, it 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 methods of locale theory are freely exploited. Third, there is substantial discussion of some computer science applications. Although books on topology aimed at mathematics exist, no book has been written specifically for computer scientists. 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.

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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 Copyright