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On the Katětov and Stone-Čech Extensions

Published online by Cambridge University Press:  20 November 2018

Bernhard Banaschewski
Bernhard Banaschewski*
Affiliation:
Hamilton College, McMaster University
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In general topology, one knows several standard extension spaces defined for one class of spaces or another and it is a natural question concerning any two such extensions which are defined for the same space whether they can ever be equal to each other. In the following, this problem will be considered for the Stone-Čech compactification βE of a completely regular non-compact Hausdorff space E[4] and Katětov's maximal Hausdorff extension κE of E[5]. It will be shown that βEκE always holds or, what amounts to the same, that κE can never be compact. As an application of this it will be proved that any completely regular Hausdorff space is dense in some non-compact space in which the Stone-Weierstrass approximation theorem holds.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 2 , Issue 1 , 01 January 1959 , pp. 1 - 4
Copyright
Copyright © Canadian Mathematical Society 1959

References

1. Alexandroff, P. et Urysohn, P., Memoire sur les espaces topologiques compact, Verh. Kon. Akad. Wet. Amsterdam.Google Scholar
2. Banaschewski, B., On the Stone-Weierstrass approximation theorem, Fund. Math. XLIV(1957), 249-252.Google Scholar
3. Bourbaki, N., Topologie gěněrale, (Paris, 1948).Google Scholar
4. Čech, E., On bicompact spaces, Ann. Math. 38 (1937), 823-844.Google Scholar
5. Katětov, M., Über H-abgeschlossene und bikompakte Räume, Časopis Mat. Fys. 69 (1939), 36-49.Google Scholar