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Strongly Hausdorff spaces

Strongly Hausdorff spaces Acta Mathematica Aeademiae Scientiarum Hungaricae Tomus 25 (3-4), (1974), pp. 245--248. By J. R. PORTER (Lawrence) 1. Introduction. In [H J, J] HAJNAL and JuH~-SZ introduced a separation axiom strictly between Hausdorff and Urysohn, called strongly Hausdorff, and used it to investigate the cardinality of discrete subsets of Hausdorff spaces. In response to a remark in [J], a simple example of a Hausdorff space which is not strongly Hausdorff is provided in Section 2 of this paper. Since an Urysohn, minimal Hausdorff space is compact, it is natural to inquire whether a strongly Hausdorff, minimal Hausdorff space is compact. A negative answer is provided in Section 3; also, in Section 3, minimal strongly Hausdorff (resp. strongly Hausdorff-closed) spaces are characterized as strongly Hausdorff spaces which are minimal Hausdorff (resp. H-closed). A Hausdorff space X is defined to be strongly Hausdotff if for each infinite subset Ac=X, there is a sequence {U~:n(N} of pairwise disjoint open sets such that A ~ U, ~ O for each n ~ N. A space is Urysohn if every pair of distinct points have disjoint closed neighbourhoods. The symbols N and Q are used to denote the set of positive integers and rational numbers, http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematica Academiae Scientiarum Hungarica Springer Journals

Strongly Hausdorff spaces

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Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general
ISSN
0001-5954
eISSN
1588-2632
DOI
10.1007/BF01886080
Publisher site
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Abstract

Acta Mathematica Aeademiae Scientiarum Hungaricae Tomus 25 (3-4), (1974), pp. 245--248. By J. R. PORTER (Lawrence) 1. Introduction. In [H J, J] HAJNAL and JuH~-SZ introduced a separation axiom strictly between Hausdorff and Urysohn, called strongly Hausdorff, and used it to investigate the cardinality of discrete subsets of Hausdorff spaces. In response to a remark in [J], a simple example of a Hausdorff space which is not strongly Hausdorff is provided in Section 2 of this paper. Since an Urysohn, minimal Hausdorff space is compact, it is natural to inquire whether a strongly Hausdorff, minimal Hausdorff space is compact. A negative answer is provided in Section 3; also, in Section 3, minimal strongly Hausdorff (resp. strongly Hausdorff-closed) spaces are characterized as strongly Hausdorff spaces which are minimal Hausdorff (resp. H-closed). A Hausdorff space X is defined to be strongly Hausdotff if for each infinite subset Ac=X, there is a sequence {U~:n(N} of pairwise disjoint open sets such that A ~ U, ~ O for each n ~ N. A space is Urysohn if every pair of distinct points have disjoint closed neighbourhoods. The symbols N and Q are used to denote the set of positive integers and rational numbers,

Journal

Acta Mathematica Academiae Scientiarum Hungarica – Springer Journals

Published: Sep 1, 1974

References

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    Bourbaki, N.
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    Hajnal, A.; Juhász, I.
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    Iliadis, S.; Fomin, S.
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    Katětov, M.
  • OnH-closed and minimal Hausdorff spaces
    Porter, J.; Thomas, J.
  • General Topology
    Willard, S.