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On Homeomorphisms between Extension Spaces

  • Bernhard Banaschewski (a1)

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In this note, conditions are obtained which will ensure that two topological spaces are homeomorphic when they have homeomorphic extension spaces of a certain kind. To discuss this topic in suitably general terms, an unspecified extension procedure, assumed to be applicable to some class of topological spaces, is considered first, and it is shown that simple conditions imposed on the extension procedure and its domain of operation easily lead to a condition of the desired kind. After the general result has been established it is shown to be applicable to a number of particular extensions, such as the Stone-Čech compactification and the Hewitt Q-extension of a completely regular Hausdorff space, Katětov's maximal Hausdorff-closed extension of a Hausdorff space, the maximal zero-dimensional compactification of a zero-dimensional space, the maximal Hausdorff-minimal extension of a semi-regular space, and Freudenthal's compactification of a rim-compact space. The case of the Hewitt Q-extension was first discussed by Heider (6).

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References

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1. Banaschewski, B., Ueber nulldimensionale Ràume, Mathematische Nachrichten, 13 (1955), 129140.
2. Banaschewski, B., Ueber zwei Extremaleigenschajten topologischer Ràume, Mathematische Nachrichten, 13 (1955), 141150.
3. Banaschewski, B. , Report of the Summer Research Institute of the Canadian Mathematical Congress, 1957.
4. Bourbaki, N., Topologie générale, Actualities scientifiques et industrielles (Paris, Hermann et Cie.).
5. Freudeuthal, H., Neuaufbau der Endentheorie, Ann. Math., 43 (1942), 261279.
6. Heider, L.J., A note concerning completely regular Gδ-spaces, Proc. Amer. Math. Soc, 8 (1957), 10601066.
7. Katetov, M., Ueber H-abgeschlossene und bikompakte Ràume, Casopis mat. a fys., 69 (1939-40), 3649.
8. Morita, K., On bicompactifications of semibicompact spaces, Sci. Rep. Tokyo Dunrika Dagaiku, Sect. A., 4 (1952), 222229.
9. Wallman, H., Lattices and topological spaces. Ann. Math., 39 (1938), 112120.
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On Homeomorphisms between Extension Spaces

  • Bernhard Banaschewski (a1)

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