Locally finite families, completely separated sets and remote points

Authors:
M. Henriksen and T. J. Peters

Journal:
Proc. Amer. Math. Soc. **103** (1988), 989-995

MSC:
Primary 54D40; Secondary 03E05

MathSciNet review:
947695

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Abstract: It is shown that if is a nonpseudocompact space with a -locally finite -base, then has remote points. Within the class of spaces possessing a -locally finite -base, this result extends the work of Chae and Smith, because their work utilized normality to achieve complete separation. It provides spaces which have remote points, where the spaces do not satisfy the conditions required in the previous works by Dow, by van Douwen, by van Mill, or by Peters.

The lemma: "Let be a space and let be a locally finite family of cozero sets of . Let be a family of zero sets of such that for each . Then is completely separated from ", is a fundamental tool in this work.

An example is given which demonstrates the value of this tool. The example also refutes an appealing conjecture--a conjecture for which the authors found that there existed significant confusion within the topological community as to its truth or falsity.

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DOI:
https://doi.org/10.1090/S0002-9939-1988-0947695-6

Article copyright:
© Copyright 1988
American Mathematical Society