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D. Rudd (1975)
A note on zero-sets in the Stone-Čech compactificationBulletin of the Australian Mathematical Society, 12
A. Makkouk (1974)
On pseudocompact and perfectly normal spacesActa Mathematica Academiae Scientiarum Hungarica, 25
L. Gillman (1961)
Rings of continuous functions
J. L. Kelley (1955)
General Topology
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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