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. 1951 On countably paracompact spaces Canad. J. Math. 3 219 – 224 10.4153/CJM-1951-026-2 . [4] Engelking , R
Abstract
We investigate the relations between decreasing sequences of sets and the insertion of semi-continuous functions, and give some characterizations of countably metacompact spaces, countably paracompact spaces, monotonically countably paracompact spaces (MCP), monotonically countably metacompact spaces (MCM), perfectly normal spaces and stratifiable spaces.
Abstract
We show that a monotonically normal space X is paracompact if and only if for every increasing open cover {U α : α < κ} of X, there is a closed cover {F nα : n < ω, α < κ} of X such that F nα ⊂ U α for n < ω, α < κ and F nα ⊂ F nβ if α ≦ β.
Abstract
We show that separable, locally compact spaces with property (a) necessarily have countable extent — i.e., have no uncountable closed, discrete subspaces — if the effective weak diamond principle ⋄(ω,ω,<) holds. If the stronger, non-effective, diamond principle Φ(ω,ω,<) holds then separable, countably paracompact spaces also have countable extent. We also give a short proof that the latter principle implies there are no small dominating families in ω 1 ω.
Abstract
A sufficient condition for the strict insertion of a continuous function between two comparable upper and lower semicontinuous functions on a normal space is given. Among immediate corollaries are the classical insertion theorems of Michael and Dowker. Our insertion lemma also provides purely topological proofs of some standard results on closed subsets of normal spaces which normally depend upon uniform convergence of series of continuous functions. We also establish a Tietze-type extension theorem characterizing closed G δ -sets in a normal space.
