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.

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

R13: Mixed Suprema

R14: Uniformities and Normal Bases

R15: S-Maps

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

R21: Order Compactifications

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

R32: Extensions of Auto-homeomorphisms

R33: Normal Bases for Finite Point Compactifications

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