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Coverings, networks and weak topologies. (English) Zbl 1138.46012

The authors continue their investigation of the topology of Banach spaces endowed with the weak topology ${E}_{w}$ and spaces of continuous functions on compact spaces with the pointwise topology ${C}_{p}\left(K\right)$. We indicate some of the most representative results.

A family of sets is called relatively open if each element is open in its union, and $\sigma$-relatively open if it is the countable union of relatively open families. A topological space is called $\sigma$-relatively metacompact if every open cover has a point-finite $\sigma$-relatively open refinement. This property is also known as “weakly $\theta$-refinable" or “weakly submetacompact" in the literature. It was asked by Hansell whether all spaces ${E}_{w}$ and ${C}_{p}\left(K\right)$ are $\sigma$-relatively metacompact. The authors already proved in [Proc. Lond. Math. Soc. (3) 75, No. 2, 349–368 (1997; Zbl 0886.54014)] that ${\ell }_{\infty }/{c}_{0}$ and ${C}_{p}\left({\omega }^{*}\right)$ fail this property under CH. In the paper under review, they improve this result by showing, within ZFC, that ${\ell }_{\infty }$ and ${C}_{p}\left(\beta \omega \right)$ fail this property.

Apart from these examples, the notions of thickly covered and $t$-metrizable space are introduced and studied. A topological space $X$ is called thickly covered if, for every open cover $ℒ$ of $X$, there exists a function $L:{\left[X\right]}^{<\omega }\to {2}^{X}$ assigning a finite union of elements of $ℒ$ to every finite subset of $X$, in such a way that $\overline{A}\subset \bigcup \left\{L\left(H\right):H\in {\left[A\right]}^{<\omega }\right\}$ for every $A\subset X$. A topological space $S$ is called $t$-metrizable if there exists a finer metrizable topology $\pi$ on $X$ and a function $J:{\left[X\right]}^{<\omega }\to {\left[X\right]}^{<\omega }$ such that, for every $A\subset X$,

$\overline{A}\subset {\overline{{\bigcup }_{H\in {\left[A\right]}^{<\omega }}J\left(H\right)}}^{\pi }·$

It is shown that all spaces ${E}_{w}$ and ${C}_{p}\left(K\right)$ are $t$-metrizable and that $t$-metrizable spaces are hereditarily thickly covered. From this fact, a number of important well-known properties of such sets can be derived, such as being angelic, monolithic and Baturov’s result that the Lindelöf number equals the extent.

The notion of thickly covered space is also related to Corson compacta and weakly Lindelöf determined spaces. Thus, a Banach space is weakly Lindelöf determined if and only if the dual space is hereditarily thickly covered in the weak${}^{*}$ topology, while a compact space $K$ is Corson if and only if ${K}^{2}\setminus {{\Delta }}_{K}$ is thickly covered. The class of Banach spaces whose dual is $t$-metrizable in the weak${}^{*}$ topology constitutes an intermediate class between weakly countably determined and weakly Lindelöf determined.

A family $\left\{{A}_{i}:i\in I\right\}$ of subsets of $X$ is called point-finitely expandable if there is a family of open sets $\left\{{B}_{i}:i\in I\right\}$ which is point-finite and ${A}_{i}\subset {B}_{i}$ for every $i\in I$. The class of spaces which have a $\sigma$-point-finitely expandable network is also considered. The authors show that ${C}_{p}\left(K\right)$ with $K$ Corson, and ${E}_{w}$ with $E$ weakly Lindelöf determined have such networks. Also studied are the relation of $t$-metrizability and the linking separability property – a stronger related property already studied by L. Oncina [Q. J. Math. 55, No. 1, 77–85 (2004; Zbl 1064.46016)] – with the existence of special kinds of networks, particularly in dual spaces with the weak${}^{*}$ topology.

##### MSC:
 46B26 Nonseparable Banach spaces 54C35 Function spaces (general topology) 46A50 Compactness in topological linear spaces; angelic spaces, etc. 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)