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On generalizations of C*-embedding for wallman rings

Part of: Maps and general types of spaces defined by maps

Published online by Cambridge University Press:  09 April 2009

H. L. Bentley  and
B. J. Taylor
H. L. Bentley
Affiliation:
The University of ToledoToledo Ohio 43606, USA
B. J. Taylor
Affiliation:
IBM Corporation Sylvania Ohio 43560, USA
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Abstract

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Biles (1970) has called a subring A of the ring C(X), of all real valued continuous functions on a topological space X, a Wallman ring on X whenever Z(A), the zero sets of functions belonging to A, forms a normal base on X in the sense of Frink (1964). Previously, we have related algebraic properties of a Wallman ring A to topological properties of the Wallman compactification w(Z(A)) of X determined by the normal base Z(A). Here we introduce two different generalizations of the concept of “a C*-embedded subset” and study relationships between these and topological (respectively, algebraic) properties of w(Z(A)) (respectively, A).

MSC classification

Secondary: 54C45: $C$- and $C^*$-embedding 54C40: Algebraic properties of function spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 25 , Issue 2 , March 1978 , pp. 215 - 229
Copyright
Copyright © Australian Mathematical Society 1978

References

REFERENCES

Aarts, J. M. (1968), “Every metric compactification is a Wallman type compactification”, Proceedings of the International Symposium on Topology and its Applications, 29–34.Google Scholar
Alo, R. A. and Shapiro, H. L. (1968), “Normal bases and compactifications”, Math. Annalen 175, 337–340.CrossRefGoogle Scholar
Banaschewski, B. (1963), “On Waliman's method of compactification”, Math. Nachr. 27, 105–114.CrossRefGoogle Scholar
Bentley, H. L. (1972), “Normal bases and compactifications” Proceedings of the Topology Conference, Univ. of Oklahoma, 23–37.Google Scholar
Bentley, H. L. (1972), “Some Wallman compactifications determined by retracts”, Proc. Amer. Math. Soc. 33, 587–593.CrossRefGoogle Scholar
Bentley, H. L. (1972), “Some Wallman compactifications of locally compact spaces”, Fund. Math. 75, 13–24.CrossRefGoogle Scholar
Bentley, H. L. and Taylor, B. J. (1975), “Wallman rings”, Pacific J. Math. 58, 15–35.CrossRefGoogle Scholar
Biles, C. M. (1970), “Gelfand and Wallman-type compactifications”, Pacific J. Math. 35, 267–278.CrossRefGoogle Scholar
Biles, C. M. (1970), “Wallman-type compactifications”, Proc. Amer. Math. Soc. 25, 363–368.CrossRefGoogle Scholar
Brooks, R. M. (1967), “On Wallman compactifications”, Fund. Math. 60, 157–173.CrossRefGoogle Scholar
Engelking, R. (1968), Outline of General Topology (Panstwowe Wydawnictwo Naukowe, Warsaw).Google Scholar
Frink, O. (1964), “Compactifications and semi-normal spaces”, Amer. J. Math. 86, 602–607.CrossRefGoogle Scholar
Giliman, L. and Jerison, M. (1960), Rings of Continuous Functions (Van Nostrand, Princeton).CrossRefGoogle Scholar
Hager, A. W. (1969), “On inverse-closed subalgebras of C(X)”, Proc. London Math. Soc. (3) 19, 233–257.CrossRefGoogle Scholar
Hager, A. W. and Johnson, D. G. (1968), “A note on certain subalgebras of C(X)”, Canad. J. Math. 20, 389–393.CrossRefGoogle Scholar
Hamburger, P. (1971), “On Wallman-type, regular Wallman-type and Z-compactifications”, Periodica Mathematica Hungarica (1) 4, 303–309.CrossRefGoogle Scholar
Henricksen, M. and Isbell, J. (1962), “Lattice ordered rings and function rings”, Pacific J. Math, 12, 533–565.CrossRefGoogle Scholar
Isbell, J. R. (1958), “Algebras of uniformly continuous functions”, Ann. Math. 68, 96–125.CrossRefGoogle Scholar
Mrówka, S. G. (1964), “Some approximation theorems for rings of unbounded functions”. Amer. Math. Soc. Not. 11, 666.Google Scholar
Naimpally, S. A. and Warrack, D. B. (1970), Proximity Spaces (Cambridge University Press, London).Google Scholar
Njåstad, O. (1966), “On Wallman-type compactifications”, Math. Zeitschr. 91, 267–276.CrossRefGoogle Scholar
Šanin, N. A. (1943), “On special extensions of topological spaces”, Dokl. Akad. Nauk S.S.S.R. 38, 6–9.Google Scholar
Simmons, G. F. (1963), Introduction to Topology and Modern Analysis (McGraw-Hill, New York).Google Scholar
Smirnov, Y. M. (1952), “Mappings of systems of open sets”, Mat. Sbornik 31 (73), 152–166.Google Scholar
Steiner, E. F. (1968), “Wallman spaces and compactifications”, Fund. Math. 61, 295–304.CrossRefGoogle Scholar
Steiner, A. K. and Steiner, E. F. (1968), “Products of compact metric spaces are regular Wailman”, Indag. Math. 30, 428–430.CrossRefGoogle Scholar
Steiner, A. K. and Steiner, E. E. (1968), “Wallman and Z-compactifications”, Duke Math. J. 35, 269–276.CrossRefGoogle Scholar
Stone, M. H. (1962), “A generalized Weierstrass approximation theorem”, Studies in Modern Analysis, M.A.A. Studies in Mathematics, Vol. I (Prentice Hall, Englewood Cliffs, N.J.).Google Scholar
Taìmanov, A. (1952), “On extension of continuous mappings of topological spaces”, Mat. Sbornik 31, 459–463.Google Scholar
Wallman, H. (1938), “Lattices and topological spaces”, Ann. of Math. 39, 112–126.CrossRefGoogle Scholar