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Sets with countable base of neighbourhoods

Sets with countable base of neighbourhoods Acta Mathematica Academiae Scientiarum Hungaricae Tomus 37 (4), (1981), 381--382. By R. MISRA (Rio de Janeiro) In this note we identify closed sets of a perfectly normal space having a countable base of neighbourhoods. The theorem leads to several useful corollaries and we ask some interesting questions. For background material and notations the reader is referred to [1]. As usual the spaces considered are completely regular and Hausdorff and .~px denotes the closure of the subset A of a space X in fiX, the Stone--l~ech compactification of X. TheOREM. Let A be a closed subset of a perfectly normal space )2. Then A has a countable base of neighbourhoods in X if and only if ]~x is a zero-set in fiX. PROOF. Necessity. Let {N~}~CN be a countable base of closed neighbourhoods of A. Perfect normality of X implies that each N~ is a zero-set neighbourhood of A and hence Ng x is a neighbourhood of ]px for each iEN. This can be seen by using [1, 7.14]. Let N be a neighbourhood of ,Tpx in fiX. Without loss of generality we can assume that N is closed in fiX. Since NN X is a neighbourhood of A, there http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematica Academiae Scientiarum Hungarica Springer Journals

Sets with countable base of neighbourhoods

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Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general
ISSN
0001-5954
eISSN
1588-2632
DOI
10.1007/BF01895137
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Abstract

Acta Mathematica Academiae Scientiarum Hungaricae Tomus 37 (4), (1981), 381--382. By R. MISRA (Rio de Janeiro) In this note we identify closed sets of a perfectly normal space having a countable base of neighbourhoods. The theorem leads to several useful corollaries and we ask some interesting questions. For background material and notations the reader is referred to [1]. As usual the spaces considered are completely regular and Hausdorff and .~px denotes the closure of the subset A of a space X in fiX, the Stone--l~ech compactification of X. TheOREM. Let A be a closed subset of a perfectly normal space )2. Then A has a countable base of neighbourhoods in X if and only if ]~x is a zero-set in fiX. PROOF. Necessity. Let {N~}~CN be a countable base of closed neighbourhoods of A. Perfect normality of X implies that each N~ is a zero-set neighbourhood of A and hence Ng x is a neighbourhood of ]px for each iEN. This can be seen by using [1, 7.14]. Let N be a neighbourhood of ,Tpx in fiX. Without loss of generality we can assume that N is closed in fiX. Since NN X is a neighbourhood of A, there

Journal

Acta Mathematica Academiae Scientiarum Hungarica – Springer Journals

Published: Jun 18, 2005

References

  • Rings of continuous functions
    Gillman, L.; Jerison, M.
  • General Topology
    Kelley, J. L.