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Normal, Locally Compact, Boundedly Metacompact Spaces are Paracompact: an Application of Pixley-Roy Spaces
Published online by Cambridge University Press: 20 November 2018
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Let PR(X) denote the Pixley-Roy topology on the collection of all nonempty, finite subsets of a space X. For each cardinal κ, let κ* be the cardinal κ with the co-finite topology. We use PR(κ*) to obtain a partial solution in ZFC to F. Tall's question whether every normal, locally compact, metacompact space is paracompact [6]. W.S. Watson has answered this question affirmatively assuming V = L[7]. The question also has an affirmative answer if we assume either that the space is perfectly normal [1] or that it is locally connected [4].
A space X is said to be boundedly metacompact (boundedly paracompact) provided that for each open cover of X there is a positive integer n such that has a point finite (locally finite) open refinement of order n. As the main result of this paper, we show every normal, locally compact, boundedly metacompact space is paracompact.
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- Copyright © Canadian Mathematical Society 1983
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