zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 (K). 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 σ-relatively open if it is the countable union of relatively open families. A topological space is called σ-relatively metacompact if every open cover has a point-finite σ-relatively open refinement. This property is also known as “weakly θ-refinable” or “weakly submetacompact” in the literature. It was asked by Hansell whether all spaces E w and C p (K) are σ-relatively metacompact. The authors already proved in [Proc. Lond. Math. Soc. (3) 75, No. 2, 349–368 (1997; Zbl 0886.54014)] that /c 0 and C p (ω * ) fail this property under CH. In the paper under review, they improve this result by showing, within ZFC, that and C p (βω) 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 of X, there exists a function L:[X] <ω 2 X assigning a finite union of elements of to every finite subset of X, in such a way that A ¯{L(H):H[A] <ω } for every AX. A topological space S is called t-metrizable if there exists a finer metrizable topology π on X and a function J:[X] <ω [X] <ω such that, for every AX,

A ¯ H[A] <ω J(H) ¯ π ·

It is shown that all spaces E w and C p (K) 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 Δ 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 {A i :iI} of subsets of X is called point-finitely expandable if there is a family of open sets {B i :iI} which is point-finite and A i B i for every iI. The class of spaces which have a σ-point-finitely expandable network is also considered. The authors show that C p (K) 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:
46B26Nonseparable Banach spaces
54C35Function spaces (general topology)
46A50Compactness in topological linear spaces; angelic spaces, etc.
54D20Noncompact covering properties (paracompact, Lindelöf, etc.)