Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-05-09T09:41:42.640Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonstandard topological extensions

Published online by Cambridge University Press:  17 April 2009

Robert A. Herrmann
Robert A. Herrmann
Affiliation:
Department of Mathematics, US Naval Academy, Annapolis, Maryland, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper investigates the nonstandard theory of filters on a non-empty meet-semi-lattice of sets and applies this theory to the general study of topological extensions Y for a space X. In particular, we apply this theory to Baire and quasi-H-closed extensions as well as Wallman type compactifications. Whereas these extensions have previously teen obtained and studied as types of ultrafilter extensions, we study them as subsets of an enlargement of X. Since X ⊂ Y ⊂ ◯ and the elements of X and Y - X are of the same set-theoretic type, these extensions appear more natural from the nonstandard viewpoint.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 13 , Issue 2 , October 1975 , pp. 269 - 290
Copyright
Copyright © Australian Mathematical Society 1975

References

[1]Alo, R.A. and Shapiro, H.L., “A note on compactificat ions and seminormal spaces”, J. Austral. Math, Soc. 8 (1968), 102–108.Google Scholar
[2]Alò, Richard A. and Shapiro, Harvey L., “Wallman compact and realcompact spaces”, Contributions to extension theory and topological structures, 9–14 (Proc. Sympos., Berlin, 196?. Deutscher Verlag der Wissenschaften, Berlin; Academic Press, New York; 1969).Google Scholar
[3]Biles, Charles M., “Wallman-type compactifications”, Proc. Amer. Math. Soc. 25 (1970), 363–368.CrossRefGoogle Scholar
[4]D'Aristotle, Anthony J., “Completely regular compactifications”, Fund. Math. 71 (1971), 139–145.Google Scholar
[5]Frink, Orrni, “Compactifications and semi-normal spaces”, Amer. J. Math. 86 (1964), 602–607.Google Scholar
[6]Gillman, Leonard and Jerison, Meyer, Rings of continuous functions (Van Nostrand, Princeton, New Jersey; Toronto; London; New York; 1960).CrossRefGoogle Scholar
[7]Herrmann, Robert A., “V-filters”, paper presented to the seventy-ninth annual meeting Amer. Math. Soc., Dallas, Texas, January, 1973 (see also: Abstract No. 701–54–14, Notices Amer. Math. Soc. 20 (1973), A–173).Google Scholar
[8]Herrmann, R.A., “H(i) semiregular, U(i) and R(i) extensions”, submitted.Google Scholar
[9]Katětov, M., “Über H-abgeschlossene und bikompakte Räume”, Časopis Pěst. Mat. Fys. 69 (1939), 36–49.Google Scholar
[10]Liu, Chen-Tung, “Absolutely closed spaces”, Trans. Amer. Math. Soc. 130 (1968), 86–104.Google Scholar
[11]Luxemburg, W.A.J., “A general theory of monads”, Applications of model theory to algebra, analysis, and probability, 18–86 (Internat. Sympos., Pasadena, California, 1967. Holt, Rinehard and Winston, New York, Chicago, San Francisco, Atlanta, Dallas, Montreal, Toronto, London, Sydney, 1969).Google Scholar
[12]Machover, Moshé, Hirschfeld, Joram, Lectures on non-standard analysis (Lecture Notes in Mathematics, 94. Springer-Verlag, Berlin, Heidelberg, New York, 1969).CrossRefGoogle Scholar
[13]McCoy, R.A., “A Baire space extension”, Proc. Amer. Math. Soc. 33 (1972), 199–202.Google Scholar
[74]Porter, Jack and Thomas, John, “On H-closed and minimal Hausdorff spaces”, Trans. Amer. Math. Soc. 138 (1969), 159–170.Google Scholar
[15]Robinson, Abraham. Non-standard analysis (Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1966).Google Scholar
[16]Samuel, P., “Ultrafilters and compactifications of uniform spaces”, Trans. Amer. Math. Soc. 64 (1948), 100–132.Google Scholar
[17]Steiner, A.K. and Steiner, E.F., “Wallman and Z-compactifications”, Duke Math. J. 35 (1968), 269–275.CrossRefGoogle Scholar
[18]Steiner, E.F., “Normal families and completely regular spaces”, Duke Math. J. 33 (1966), 743–745.Google Scholar
[19]Steiner, E.F., “Wallman spaces and compactifications”, Fund. Math. 61 (1967/1968), 295–304.Google Scholar