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Wallman compactifications and Wallman realcompactifications

Published online by Cambridge University Press:  09 April 2009

M. S. Gagrat  and
S. A. Naimpally
M. S. Gagrat
Affiliation:
Mathematics DepartmentIndian Institute of Technology, Kanpur Kanpur–16, U.P. India
S. A. Naimpally
Affiliation:
Mathematics DepartmentIndian Institute of Technology, Kanpur Kanpur–16, U.P. India
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In 1964 Frink [8] generalized Wallman's method [17] of compactification and asked the question: “Is every Hausdorff compactification of a Tychonoff space a Wallman compactification?”. This problem, which is as yet unsolved, has led to the discovery of a number of necessary and/or sufficient conditions for a Hausdorff compactification to be Wallman; (see Alò and Shapiro [2], [3], Banasĉhewski [6], Njåstad [13] and Steiner [15]). Recently Alò and Shapiro [4], [5] have generalized the Wallman procedure to discuss what they call Z*-realcompact spaces and Z*-realcompactification n(Z) corresponding to a countably productive (c.p.) normal base Z on X. Some further work has been done by Steiner and Steiner [14].

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 15 , Issue 4 , June 1973 , pp. 417 - 427
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Aarts, J. M., Dimension and deficiency in topology (Ph.D.thesis, Amsterdam, 1966).Google Scholar
[2]Alò, R.A. and Shapiro, H. L., ‘A note on compactifications and semi-normal spaces’, J. Austral. Math. Soc. 8 (1968), 102–108.CrossRefGoogle Scholar
[3]Alò, R. A. and Shapiro, H. L., ‘Normal bases and compactifications’, Math. Annalen 175 (1968), 337–340.CrossRefGoogle Scholar
[4]Alò, R.A. and Shapiro, H.L., ‘Z-realcompactifications and normal bases’, J. Austral. Math. Soc. 9 (1969), 489–495.Google Scholar
[5]Alò, R. A. and Shapiro, H. L., ‘Wallman compact and realcompact spaces’, Contr. Ext. Theory of Topological Structures, Berlin (1969), 9–14.Google Scholar
[6]Banasčhewski, B., ‘On Wallman's method of compactification’, Math. Nachricten 27 (1963), 105–114.CrossRefGoogle Scholar
[7]Efremovič, V. A., ‘The geometry of proximity I,’ Mat. Sb. 31(73) (1952), 189–200.Google Scholar
[8]Frink, O., ‘Compactifications and semi-normal spaces’, Amer. J. Math. 86 (1964), 602–607.CrossRefGoogle Scholar
[9]Gagrat, M. S. and Naimpally, S.A., ‘Proximity approach to extension problems’, Fund. Math. 71 (1966), 63–76.Google Scholar
[10]Gillman, L. and Jerison, M., Rings of continuous functions (Van Nostrand, Princeton, 1960).CrossRefGoogle Scholar
[11]Lodato, M. W., ‘On topologically induced generalized proximity relations II’, Pacific J. Math. 17 (1966), 131–135.CrossRefGoogle Scholar
[12]Mrowka, S., ‘Some properties of Q-spaces’, Bull. Acad. Polon. Sci. 5 (1957), 947–950.Google Scholar
[13]Njåstad, O., ‘On Wallman-type compactifications’, Math. Z. 91 (1966), 267–276.CrossRefGoogle Scholar
[14]Steiner, A.K. and Steiner, E.F. ‘Nest generated intersections rings in Tychonoff spaces’, Trans. Amer. Math. Soc., 148 (1970), 589–601.Google Scholar
[15]Steiner, E. F., ‘Wallman spaces and compactifications’, Fund. Math. 61 (1968), 295–304.Google Scholar
[16]Steiner, A. K. and Steiner, E. F., ‘Wallman and Z-compactifications’, Duke Math. J. 35 (1968), 269–276.Google Scholar
[17]Wallman, H., ‘Lattices and topological spaces’, Ann. of Math. 39 (1938), 112–126.Google Scholar