Foundations of topology : an approach to convenient topology / by Gerhard Preuss.

Format:

Book

Author/Creator:

Preuss, Gerhard, 1940-

Contributor:

Alumni and Friends Memorial Book Fund.

Language:

English

Subjects (All):

Topology.

Physical Description:

xviii, 303 pages : illustrations ; 25 cm

Place of Publication:

Dordrecht ; Boston : Kluwer Academic Publishers, [2002]

Contents:

0.1 Set theoretical concepts 1

0.2 Topological structures 8

0.2.1 Metric and pseudometric spaces 8

0.2.2 Topological spaces and continuous maps 10

0.2.3 Filters and convergence 13

0.2.4 Initial and final topological structures 17

0.3 Some categorical concepts 22

Chapter 1. Topological constructs 29

1.2 Special categorical properties of topological constructs 34

1.2.1 Completeness and cocompleteness 34

1.2.2 Special objects and special morphisms 38

Chapter 2. Reflections and coreflections 45

2.1 Universal maps and adjoint functors 46

2.2 Characterization theorems of E-reflective and M-coreflective subcategories 58

2.3 Examples of bireflections and bicoreflections 67

2.3.1 Convergence structures 67

2.3.2 Uniform convergence structures 72

2.3.3 The linkage between convergence structures and uniform convergence structures 78

Chapter 3. Topological universes 91

3.1 Cartesian closed topological constructs 92

3.2 Extensional topological constructs 102

3.3 Strong topological universes 108

Chapter 4. Completions of semiuniform convergence spaces 111

4.1 Completion of uniform spaces 112

4.2 Regular completion of semiuniform convergence spaces 115

4.2.1 Regular spaces 115

4.2.2 Regular completion via function spaces 120

4.3 Applications to compactifications 129

4.3.1 An alternative description of uniform spaces by means of uniform covers 129

4.3.2 Precompactness and compactness 133

4.3.3 Hausdorff compactifications of topological spaces 142

4.4 The simple completion and the Wyler completion 146

Chapter 5. Connectedness properties 161

5.2 Disconnectednesses and their relations to connectednesses 170

5.3 Local E-connectedness 176

Chapter 6. Function spaces 181

6.1 Simple convergence, uniform convergence and continuous convergence in the realm of classical General Topology 182

6.2 Local compactness and local precompactness in semiuniform convergence spaces 193

6.3 Precompactness and compactness in the natural function spaces of the construct of semiuniform convergence spaces 206

Chapter 7. Relations between semiuniform convergence spaces and merotopic spaces (including nearness spaces) 219

7.1 An alternative description of filter spaces in the realm of merotopic spaces 221

7.2 Subtopological spaces 225

7.3 Complete regularity and normality 231

7.4 Paracompactness and dimension 240

7.5 Subcompact and sub-(compact Hausdorff) spaces 248

Appendix. Some algebraically topological aspects in the realm of Convenient Topology 255

A.1 Cohomology for filter spaces 255

A.2 Path connectedness and fundamental groups for limit spaces 260

Implication scheme for various SUConv-invariants 273

Table: Preservation properties of some SUConv-invariants 277

Diagram of relations between various subconstructs of SUConv (including their relations to merotopic and nearness spaces) 279.

Notes:

Includes bibliographical references (pages [281]-289) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Alumni and Friends Memorial Book Fund.

ISBN:

1402008910

OCLC:

50479364