Extensions and Absolutes of Hausdorff Spaces

An extension of a topological space X is a space that contains X as a dense subspace. The construction of extensions of various sorts - compactifications, realcompactifications, H-elosed extension- has long been a major area of study in general topology. A ubiquitous method of constructing an extension of a space is to let the "new points" of the extension be ultrafilters on certain lattices associated with the space. Examples of such lattices are the lattice of open sets, the lattice of zero-sets, and the lattice of elopen sets. A less well-known construction in general topology is the "absolute" of a space. Associated with each Hausdorff space X is an extremally disconnected zero-dimensional Hausdorff space EX, called the Iliama absolute of X, and a perfect, irreducible, a-continuous surjection from EX onto X. A detailed discussion of the importance of the absolute in the study of topology and its applications appears at the beginning of Chapter 6. What concerns us here is that in most constructions of the absolute, the points of EX are certain ultrafilters on lattices associated with X. Thus extensions and absolutes, although very different, are constructed using similar tools.

Table of Contents:
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.- 6Extremally 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.