Geodesic
Normal, Locally Compact, Boundedly Metacompact Spaces are Paracompact: an Application of Pixley-Roy Spaces
Canadian journal of mathematics, Tome 35 (1983) no. 5, pp. 807-823

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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. 
Daniels, Peg. Normal, Locally Compact, Boundedly Metacompact Spaces are Paracompact: an Application of Pixley-Roy Spaces. Canadian journal of mathematics, Tome 35 (1983) no. 5, pp. 807-823. doi: 10.4153/CJM-1983-046-x
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