Network characterization of Gul'ko compact spaces and their relatives

Abstract

In this paper we characterize the classes of Gul'ko and Talagrand compact spaces through a network condition leading us to answer two questions posed by G. Gruenhage [Proc. Amer. Math. Soc. 100 (1987) 371-376] on covering properties.

Point Networks for Special Subspaces of $\mathbb{R}^κ$

“…The proof is essentially the same as that given by Garcia et al in [8] for the case when P = K(M ), which in turn was a straightforward extension of Gruenhage's original argument for the P = N case. Consequently we only sketch the proof for general directed sets P .…”

“…Consequently we only sketch the proof for general directed sets P . We start with a lemma which is the P -analogue of Proposition 8 of [8].…”

P-paracompact and P-metrizable spaces

“…By analogy with the property 'weakly σ-point finite' used in [3], we call a family C of subsets of a space X weakly σ-locally finite if we can write C = n C n in such a way that: A space Y is a ∆-space if whenever we write Y as an increasing union of subsets, Y = n S n where S n ⊆ S n+1 for all n, there is a countable closed cover, {C n : n ∈ ω}, of Y , such that C n ⊆ S n for every n. A subset A of R is a ∆-set if it is an uncountable ∆-space. It is clear that every Q-space is a ∆-space.…”

“…By analogy with the property 'weakly σ-point finite' used in [3], we call a family C of subsets of a space X weakly σ-locally finite if we can write C = n C n in such a way that:…”

“…Indeed a compact space X is (i) Eberlein, (ii) Talagrand, or (iii) Gul'ko if and only if X 2 \ ∆ is (i) ′ Nmetacompact, (ii) ′ K(ω ω )-metacompact, or (iii) ′ K(M )-metacompact, for some separable metrizable M (respectively). The equivalence of (i) and (i) ′ is due to Gruenhage, [7], while the other two equivalences are due to Garcia, Orihuela and Oncina, [3]. The first two authors gave in [2] a systematic and uniform development of the theory of those compact spaces X such that X 2 \ ∆ is P -metacompact (P -Eberlein compact), giving alternative characterizations in terms of almost subbases, bases, networks and point networks.…”