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Non-Hausdorff Topology and Domain Theory

Book Description

This unique book on modern topology looks well beyond traditional treatises and explores spaces that may, but need not, be Hausdorff. This is essential for domain theory, the cornerstone of semantics of computer languages, where the Scott topology is almost never Hausdorff. For the first time in a single volume, this book covers basic material on metric and topological spaces, advanced material on complete partial orders, Stone duality, stable compactness, quasi-metric spaces and much more. An early chapter on metric spaces serves as an invitation to the topic (continuity, limits, compactness, completeness) and forms a complete introductory course by itself. Graduate students and researchers alike will enjoy exploring this treasure trove of results. Full proofs are given, as well as motivating ideas, clear explanations, illuminating examples, application exercises and some more challenging problems for more advanced readers.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Contents
  6. 1. Introduction
  7. 2. Elements of set theory
    1. 2.1 Foundations
    2. 2.2 Finiteness, countability
    3. 2.3 Order theory
    4. 2.4 The Axiom of Choice
  8. 3. A first tour of topology: metric spaces
    1. 3.1 Metric spaces
    2. 3.2 Convergence, limits
    3. 3.3 Compact subsets
    4. 3.4 Complete metric spaces
    5. 3.5 Continuous functions
  9. 4. Topology
    1. 4.1 Topology, topological spaces
    2. 4.2 Order and topology
    3. 4.3 Continuity
    4. 4.4 Compactness
    5. 4.5 Products
    6. 4.6 Coproducts
    7. 4.7 Convergence and limits
    8. 4.8 Local compactness
    9. 4.9 Subspaces
    10. 4.10 Homeomorphisms, embeddings, quotients, retracts
    11. 4.11 Connectedness
    12. 4.12 A bit of category theory I
  10. 5. Approximation and function spaces
    1. 5.1 The way-below relation
    2. 5.2 The lattice of open subsets of a space
    3. 5.3 Spaces of continuous maps
    4. 5.4 The exponential topology
    5. 5.5 A bit of category theory II
    6. 5.6 C-generated spaces
    7. 5.7 bc-domains
  11. 6. Metrics, quasi-metrics, hemi-metrics
    1. 6.1 Metrics, hemi-metrics, and open balls
    2. 6.2 Continuous and Lipschitz maps
    3. 6.3 Topological equivalence, hemi-metrizability, metrizability
    4. 6.4 Coproducts, quotients
    5. 6.5 Products, subspaces
    6. 6.6 Function spaces
    7. 6.7 Compactness and symcompactness
  12. 7. Completeness
    1. 7.1 Limits, d-limits, and Cauchy nets
    2. 7.2 A strong form of completeness: Smyth-completeness
    3. 7.3 Formal balls
    4. 7.4 A weak form of completeness: Yoneda-completeness
    5. 7.5 The formal ball completion
    6. 7.6 Choquet-completeness
    7. 7.7 Polish spaces
  13. 8. Sober spaces
    1. 8.1 Frames and Stone duality
    2. 8.2 Sober spaces and sobrification
    3. 8.3 The Hofmann–Mislove Theorem
    4. 8.4 Colimits and limits of sober spaces
  14. 9. Stably compact spaces and compact pospaces
    1. 9.1 Stably locally compact spaces, stably compact spaces
    2. 9.2 Coproducts and retracts of stably compact spaces
    3. 9.3 Products and subspaces of stably compact spaces
    4. 9.4 Patch-continuous, perfect maps
    5. 9.5 Spectral spaces
    6. 9.6 Bifinite domains
    7. 9.7 Noetherian spaces
  15. References
  16. Notation index
  17. Index