Author | Porter, Jack R. author |
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Title | Extensions and Absolutes of Hausdorff Spaces [electronic resource] / by Jack R. Porter, R. Grant Woods |
Imprint | New York, NY : Springer New York, 1988 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-3712-9 |
Descript | XIII, 856p. 27 illus. online resource |
1 Topological background -- 1.1 Notation and terminology from elementary set theory -- 1.2 Notation and terminology for elementary topological concepts -- 1.3 C(X) as a lattice-ordered ring -- 1.4 Tychonoff spaces, zero-sets, and cozero-sets -- 1.5 Clopen sets and zero-dimensional spaces -- 1.6 Continuous functions -- 1.7 Product spaces and evaluation maps -- 1.8 Perfect functions -- 1.9 C- and C*-embedding -- 1.10 Normal spaces -- 1.11 Pseudocompact spaces -- Problems -- 2 Lattices, filters, and topological spaces -- 2.1 Posets and lattices -- 2.2 Regular open sets, regular closed sets, and semiregular spaces -- 2.3 Filters on a lattice -- 2.4 More lattice properties -- 2.5 Completions of lattices and ordered topological spaces -- 2.6 Ordinals, cardinals, and spaces of ordinals -- Problems -- 3 Boolean algebras -- 3.1 Definition and basic properties -- 3.2 Stoneโs representation and duality theorems -- 3.3 Atomless, countable Boolean algebras -- 3.4 Completions of Boolean algebras -- 3.5 The continuum hypothesis and Martinโs Axiom -- Problems -- 4 Extensions of spaces -- 4.1 Basic concepts -- 4.2 Compactifications -- 4.3 One-point compactifications -- 4.4 Wallman compactifications -- 4.5 Gelfand compactifications -- 4.6 The Stone-?ech compactification -- 4.7 Zero-dimensional compactifications -- 4.8 H-closed spaces -- Problems -- 5 Maximum P-extensions -- 5.1 Introductory remarks -- 5.2 P-regular and P-compact spaces -- 5.3 Characterizations of extension properties -- 5.4 E-compact spaces -- 5.5 Examples of E-compactness -- 5.6 Tychonoff extension properties -- 5.7 Zero-dimensional extension properties -- 5.8 Hausdorff extension properties -- 5.9 More on Tychonoff and zero-dimensional extension properties -- 5.10 Two examples of maximum P-extensions -- 5.11 Realcompact spaces and extensions -- Problems -- 6 Extremally disconnected spaces and absolutes -- 6.1 Introduction -- 6.2 Characterization of extremally disconnected spaces -- 6.3 Examples of extremally disconnected spaces -- 6.4 Extremally disconnected spaces and zero-dimensionality -- 6.5 Irreducible functions -- 6.6 The construction of the Iliadis absolute -- 6.7 The uniqueness of the absolute -- 6.8 The construction of EX as a space of open ultrafilters -- 6.9 Elementary properties of EX -- 6.10 Examples of absolutes -- 6.11 The Banaschewski absolute -- Problems -- 7 H-closed extensions -- 7.1 Strict and simple extensions -- 7.2 The Fomin extension -- 7.3 One-point H-closed extensions -- 7.4 Partitions of ?X\X -- 7.5 Minimal Hausdorff spaces -- 7.6 p-maps -- 7.7 An equivalence relation on H(X) -- Problems -- 8 Further properties and generalization of absolutes -- 8.1 Introduction -- 8.2 Absolutes and H-closed extensions -- 8.3 Absolutes and extension properties -- 8.4 Covers of topological spaces -- 8.5 Completions of C(X) vs. C(EX) -- Problems -- 9 Categorical interpretations of absolutes and extensions -- 9.1 Introduction -- 9.2 Categories, functors, natural transformations, and subcategories -- 9.3 Topological categories -- 9.4 Morphisms -- 9.5 Products and coproducts -- 9.6 Reflective and epireflective subcategories -- 9.7 Coreflections -- 9.8 Projective covers -- Problems -- Notes -- List of Symbols