Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-24T10:32:06.346Z Has data issue: false hasContentIssue false
  • English
  • Français

General rings of functions

Published online by Cambridge University Press:  09 April 2009

A. Sultan
A. Sultan
Affiliation:
Queens CollegeFlushing, New York 11367.
Rights & Permissions [Opens in a new window]

Extract

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.

Several authors have studied various types of rings of continuous functions on Tychonoff spaces and have used them to study various types of compactifications (See for example Hager (1969), Isbell (1958), Mrowka (1973), Steiner and Steiner (1970)). However many important results and properties pertaining the Stone-Čech compactification and the Hewitt realcompactification can be extended to a more general setting by considering appropriate lattices of sets, generalizing that of the lattice of zero sets in a Tychonoff space. This program was first considered by Wallman (1938) and Alexandroff (1940) and has more recently appeared in Alo and Shapiro (1970), Banachewski (1962), Brooks (1967), Frolik (1972), Sultan (to appear) and others.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 20 , Issue 3 , November 1975 , pp. 359 - 365
Copyright
Copyright © Australian Mathematical Society 1975

References

Alexandroff, A. D. (1940), ‘Additive set functions in abstract spaces’ Mat. Sb. (N. S.) 8, (50) 307–348.Google Scholar
Alo, R. and Shapiro, H. L. (1970), ‘Z-realcompactifications and normal bases’, J. Austral. Math. Soc. 11, 489–495.Google Scholar
Banachewski, B. (1962), ‘Normal systems of sets,’ Math. Nachr. 24, 53–75.CrossRefGoogle Scholar
Brooks, R. M. (1967), ‘On Wallman compactifications,’ Fund. Math. 60, 157–173.CrossRefGoogle Scholar
Frink, O. (1964), ‘Compactifications and semi-normal spaces’ Amer. J. Math. 86, 602–607.CrossRefGoogle Scholar
Frolik, Z. (1972), ‘Prime filters with the C.I.P.,’ Comment. Math. Univ. Carolinae 13, 553–575.Google Scholar
Gillman, L. and Jerrison, M. (1960), Rings of continuous functions (D. Van Nostrand. Princeton, N.J. 1960).CrossRefGoogle Scholar
Hager, A. W., (1969), ‘On inverse closed subalgebras of C(X),’ Proc. London Math. Soc. (3) 19, 233–257.CrossRefGoogle Scholar
Isbell, J. R. (1958), ‘Algebras of uniformly continuous functions’, Ann. of Math. 68, 96–125.CrossRefGoogle Scholar
Mrowka, S. (1973), ‘β-Like compactifications’, Acta Math. Acad. Sci. Hungar. 24, 279–287.CrossRefGoogle Scholar
Steiner, A. K. and Steiner, E. F. (1970), ‘Nest generated intersection rings in Tychonoff spaces’, Trans. Amer. Math. Soc. 148, 559–601.CrossRefGoogle Scholar
Sultan, A. (to appear), ‘Lattice realcompactifications’, Ann. Mat. Pura Appl.Google Scholar
Sultan, A., ‘Generalization and unification of five approaches to the Stone-Čech compactification’, (submitted).Google Scholar
Wallman, H. (1938), ‘Lattices and topological spaces’, Ann. of Math. (2) 39, 112–126.CrossRefGoogle Scholar