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Minimal Hausdorff and maximal compact spaces

Published online by Cambridge University Press:  09 April 2009

N. Smythe  and
C. A. Wilkins
N. Smythe
Affiliation:
University of New Southy Wales.
C. A. Wilkins
Affiliation:
University of New Southy Wales.
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Given two topologies J1, J2 on a set X, J1 is said to be coarser than J2, written J1 ≦ J2, if every set open under J1 is open under J2. A minimal Hausdorff space is then one for which there is no coarser Hausdorff topology etc. Vaidyanathaswamy [4] showed that every compact Hausdorff space is both maximal compact and minimal Hausdorff. This raised the question of whether there exist minimal Hausdorff non-compact spaces and/or maximal compact non-Hausdorff spaces. These questions were in fact answered in the affirmative by Ramanathan [2], Balachandran [1], and Hing Tong [3]. Their examples were, however, all on countable sets, and the topology constructed to answer one question bore no relation to the topology answering the second. In particular, the minimal Hausdorff non-compact topologies were not finer than any maximal compact topology.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 3 , Issue 2 , May 1963 , pp. 167 - 171
Copyright
Copyright © Australian Mathematical Society 1963

References

[1]Balachandran, V. K., Minimal Bicompact Spaces. J. Ind. Math. Soc., N.S. Vol. 12, pp. 47–48 (1948).Google Scholar
[2]Ramanathan, A., A Characterisation of Maximal Hausdroff Spaces, J. Ind. Math. Soc., N.S. Vol. 11, pp. 23–80 (1947).Google Scholar
[3]Tong, H., Minimal Bicompact Spaces, Bull. Am. Math Soc., Vol. 54, pp. 478–479 (1948).Google Scholar
[4]Vaidyanathaswamy, R., Treatise on Set Topology (Ind. Math. Soc. Madras 1947).Google Scholar