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On paracompact regular spaces
Published online by Cambridge University Press: 09 April 2009
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A topological space is paracompact if and only if each open cover of the space has an open locally finite refinement. It is well-known that an unusual normality condition is satisfied by each paracompact regular space X [p. 158, 5]: Let α be a locally finite (discrete) family of subsets of X, then there is a neighborhood V of the diagonal Δ(X) (in X × X), such that V[x] intersects at most a finite number of members (respectively at most one member) of {V[A]: A ∈ α} for each x ∈ X. In this not we will show that a variant of this condition actually characterizes paracompactness. Among other results, an improvement to a recent result of H. H.Corson [2] is given so as to accord with a conjecture of J. L. Kelley [p. 208, 5] more prettily, and we connect paracompactness to metacompactness [1]
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- Copyright © Australian Mathematical Society 1961
References
[1]Arens, R. and Dungundji, J., Remarks on the concept of compactness, Portugaliae Math. 9 (1950).Google Scholar
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