Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Compactification and completion as absolute closure

Author: Anthony W. Hager
Journal: Proc. Amer. Math. Soc. 40 (1973), 635-638
MSC: Primary 54D35
MathSciNet review: 0331326
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is well known that a Tychonoff (respectively, Hausdorff uniform) space is compact (resp., complete) iff it is absolutely closed, i.e., dense in no other such space; we shall sketch a proof of this to our purpose, which is: Given $ X$, we apply Zorn's Lemma to obtain a space maximal with respect to the property of containing $ X$ densely, thus compact or complete. For uniform spaces, the ``maximal extension'' is automatically the completion; for Tychonoff spaces, we must, and do, explicitly arrange things so that the ``maximal extension'' has the universal mapping property describing the Stone-Čech compactification. A variant of the construction yields the Hewitt realcompactification. A crucial point in the proofs is (of course) the exhibition of an upper bound for a chain; this is, in essence, a direct limit construction.

References [Enhancements On Off] (What's this?)

  • [1] N. Bourbaki, Éléments de mathématique, Fasc. II. Livre III. Topologie générale. Chap. 1: Structures topologiques, Actualités Sci. Indust., no. 1142, Hermann, Paris, 1961; English transl., Hermann, Paris, Addison-Wesley, Reading, Mass., 1966. MR 34 #5044a; MR 25 #4480.
  • [2] W. W. Comfort, A theorem of Stone-Čech type, and a theorem of Tychonoff type, without the axiom of choice; and their realcompact analogues, Fund. Math 63 (1968), 97-110. MR 38 #5174. MR 0236880 (38:5174)
  • [3] J. Dugundji, Topology, Allyn and Bacon, Boston, Mass., 1966. MR 33 #1824. MR 0193606 (33:1824)
  • [4] L. Gillman and M. Jerison, Rings of continuous functions, Univ. Series in Higher Math., Van Nostrand, Princeton, N.J., 1960. MR 22 #6994. MR 0116199 (22:6994)
  • [5] D. W. Hajek and G. Strecker, Direct limits of Hausdorff spaces (to appear). MR 0355937 (50:8410)
  • [6] H. Herrlich, Separation axioms and direct limits, Canad. Math. Bull. 12 (1969), 337-338. MR 40 #6487. MR 0253272 (40:6487)
  • [7] J. R. Isbell, Uniform spaces, Math. Surveys, no. 12, Amer. Math. Soc., Providence, R.I., 1964. MR 30 #561. MR 0170323 (30:561)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54D35

Retrieve articles in all journals with MSC: 54D35

Additional Information

Keywords: Compactification, completion, absolutely closed
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society