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

R14: Uniformities and Normal Bases

R16: The Remnant Rings as Compactifications

R17: Algebraic Structure of the Remnant Rings

R18: Metrizable Compactifications

R19: Ordering the Remnant Rings

R20: p-adic Tools for the Remnant Rings

R22: Extensions and Compactification

R23: Special Cases of Extensions

R24: Disjoint Unions of Uniformities

R25: Compactifications and Hyperspaces

R26: The Remnant Rings Are Homeomorphic

R27: Normal Bases for the Remnant Rings

R28: Order-Reversing Involutions for the Remnant Rings

R29: Point Spaces of the Remnant Rings

R30: Realcompactness and Uniformity

R31: Possible Applications to Number Theory

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