Suprema of Compactifications

This is a collection of results and examples related to compactifications, with emphasis on the supremum operation. Some of them are well-known and a few may be new, while others are perhaps known but obscure. Because the author does not have access to a research library, some significant references may have been unintentionally omitted. An extensive bibliography can be found in Chandler and Faulkner's article "Hausdorff Compactifications: A Retrospective," which appears in Handbook of the History of General Topology, Volume 2. (Edited by C.E. Aull and R. Lowen, Kulwer Academic Publishers, 1998.) Brief abstracts for each section on this website can be found in the overview below.

Each heading below will link to a PDF file produced by PCTeX. A basic familiarity with topology, set theory, abstact algebra, lattice theory, and elementary number theory is assumed. Sections whose heading is labeled 'P' establish notation and state definitions and facts (without proof), all of which will be used in the other sections as needed. In the 'R' sections detailed proofs are given.

Overview
P1: Ordering of Compactifications
P2: Uniform Spaces
P3: Normal Bases
P4: Nets
R1: Existence of Suprema via Uniform Space Theory
R2: Existence of Suprema via Quotients of the Stone-Cech Compactification
R3: Representation of Suprema
R4: Suprema of Countably Infinite Families
R5: Finite-point Compactifications
R6: Suprema of Two-point Compactifications
R7: Uniform Continuity and Extension of Maps
R8: Lattice and Semi-Lattice Properties
R9: Directed Sets of Normal Bases
R10: Some Metric Compactifications of the Natural Numbers
R11: The Magill-Glasenapp Theorem
R12: Extending Arithmetic Operations