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Directed Covers and Paracompact Spaces

1967; Cambridge University Press; Volume: 19; Linguagem: Inglês

10.4153/cjm-1967-059-0

1496-4279

John Mack,

Williams Syndrome Research

A cover is directed (is well-ordered, totally ordered) if it is directed (is well-ordered, totally ordered, respectively) by set inclusion. Specifically is directed if, given U, V in , there exists W in such that U ⋃ V ⊂ W . Directed covers have been used implicitly in numerous theorems dealing with paracompactness and related properties. In particular, characterizations of paracompactness in terms of product spaces seem to require the use of directed covers (3; 11, §2; or 12). However, the utility of directed covers has not been fully exploited.