A generalization of a theorem of Aquaro

Abstract

In this paper we introduce the concept of a δθ-cover to generalize Aquaro's Theorem that every point countable open cover of a topological space such that every discrete closed family of sets is countable has a countable subcover. A δθ-cover of a space X is defined to be a family of open sets where each Vn covers X and for x є X there exists n such that Vn is of countable order at x. We replace point countable open cover by a δθ-cover in Aquaro's Theorem and also generalize the result of Worrell and Wicke that a θ-refinable countably compact space is compact and Jones′ result that ℵ1-compact Moore space is Lindelöf which was used to prove his classic result that a normal separable Moore space is metrizable, using the continuum hypothesis.

Point-Countability and Compactness

“…Then X is called weakly 89-refinable. If, in addition to (1), <¥ satisfies (3), then X is called 89-refinable [3]. If every open covering of X has an open refinement % satisfying (2), X is called weakly 0-refinable [5].…”

“…For the paracompact case this is due to Dieudonne [8], for metacompact spaces to Arens and Dugundji [2], for metaLindelof spaces to Aquaro [1], for t9-refinable spaces to Worrell and Wicke [19], and for <50-refinable spaces to Aull [3].…”

s‐point finite refinable spaces

“…A T 1 -space X is compact (a T 3 -space Lindelöf) if and only if every open cover ᐁ of X has an open refinement ᐂ which is C-point finite for some finite (countable) C ⊆ X. More generally, in [2,1] it is noted that if ᐁ is an open cover of a T 1 -space X then there is a closed discrete A ⊆ X such that ᐁ A = ∪{(ᐁ) a : a ∈ A} is a subcover of ᐁ. It is readily seen that a (T 3 -) space X is metacompact (paracompact) if and only if for every open cover ᐁ of X there is an open refinement ᐂ such that for some closed discrete set C ⊆ X, ᐂ is C-point finite and (iii) the collection {st (a, ᐂ) : a ∈ C} is point finite (locally finite).…”

A remark on irreducible spaces

“…Some applications. In [3] C. E. Aull introduced the notion of a distinguished point set and a 5f9-cover and thereby generalized a theorem of G. Aquaro [1], The following, through Theorem 3.4, are found in [3].…”