Extensions and Absolutes of Hausdorff Spaces

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 Stone's representation and duality theorems.- 3.3 Atomless, countable Boolean algebras.- 3.4 Completions of Boolean algebras.- 3.5 The continuum hypothesis and Martin's 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.