A 1-point compactification of R A 2-circle compactification of R

        A 1-point Compactification of ℝ1 Embedded in ℝ3        

        A 2-circle Compactification of ℝ1 Embedded in ℝ3        

Note: The pictures above are primarily exercises in Maple programming. The 1-point and 2-circle compactifications of ℝ1 can both be embedded in ℝ2: Compactifications of R

Hausdorff Compactifications: A Retrospective

Richard E. Chandler and Gary D. Faulkner
Handbook of the History of General Topology, Volume 2
C.E. Aull, R. Lowen, Editors
Springer - ISBN 978-0-7923-5030-9, April 1998
Hausdorff compactifications began to be studied in the 1930's and continues to be a vital topic in topology today. This paper is an historical look at the subject.

Section Headings:

0. Introduction: Compactness
1. Compactifications: Early Efforts
2. Modern Notations and Conventions
3. The Structure of K(X)
4. Special Properties
5. Singular Compactifications
6. βℕ
7. Alternative Constructions of βX
8. The Axiom of Choice
9. Wallman-Frink Compactifications
10. Secondary Sources
11. Research Levels Since 1940
Appendix (On the origin of the expression "Stone-Čech compactification")
Appendix (On R. G. Lubben's discovery of the "Stone-Čech" compactification)
References

Return to Main Page