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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
Acta Mathematica Academiae Scientiarum Hungarica – Springer Journals
Published: Jun 18, 2005
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