On the Products of Weakly Lindelof Spaces

Hajnal, A.; Juhasz, I.

doi:10.2307/2040282

Cited by 4 publications

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“…Proof. In [15] Hajnal and Juhasz give examples X and Y of Lindelöf spaces for which X × Y is not weakly Lindelöf 2 . These examples are constructed in ZFC.…”

Section: Possibilities Of Failure For Productsmentioning

confidence: 99%

Weak covering properties and selection principles

Babinkostova

Pansera

Scheepers

2013

Topology and its Applications

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No convenient internal characterization of spaces that are productively Lindelöf is known. Perhaps the best general result known is Alster's internal characterization, under the Continuum Hypothesis, of productively Lindelöf spaces which have a basis of cardinality at most ℵ 1 . It turns out that topological spaces having Alster's property are also productively weakly Lindelöf. The weakly Lindelöf spaces form a much larger class of spaces than the Lindelöf spaces. In many instances spaces having Alster's property satisfy a seemingly stronger version of Alster's property and consequently are productively X, where X is a covering property stronger than the Lindelöf property. This paper examines the question: When is it the case that a space that is productively X is also productively Y, where X and Y are covering properties related to the Lindelöf property.Problem 2 (E.A. Michael). If X is a productively Lindelöf space, then is X ℵ0 a Lindelöf space?2010 Mathematics Subject Classification. 54B10, 54D20, 54G10, 54G12, 54G20.

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“…Proof. In [15] Hajnal and Juhasz give examples X and Y of Lindelöf spaces for which X × Y is not weakly Lindelöf 2 . These examples are constructed in ZFC.…”

Section: Possibilities Of Failure For Productsmentioning

confidence: 99%

Weak covering properties and selection principles

Babinkostova

Pansera

Scheepers

2013

Topology and its Applications

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“…We will also consider the countable versions of notions and results proved in [Arc79], [BGW78], and [WD84]. For counterexamples and less known theorems we use sources such as [CS96], [SZ10],and [HJ75].…”

Section: Standard Notation Definitions and Resultsmentioning

confidence: 99%

A Comparison of Lindelof-Type Covering Properties of Topological Spaces

Staynova¹

2012

Preprint

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Lindelöf spaces are studied in any basic Topology course. However, there are other interesting covering properties with similar behaviour, such as almost Lindelöf, weakly Lindelöf, and quasi-Lindelöf, that have been considered in various research papers. Here we present a comparison between the standard results on Lindelöf spaces and analogous results for weakly and almost Lindelöf spaces. Some theorems, similar to the published ones, will be proved. We also consider counterexamples, most of which have not been included in the standard Topological textbooks, that show the interrelations between those properties and various basic topological notions, such as separability, separation axioms, first countability, and others. Some new features of those examples will be noted in view of the present comparison. We also pose several open questions. Historical Overview and MotivationOne of the basic theorems in Real Analysis, the Heine-Borel Theorem, states (in modern terminology) that every closed interval on the real line is compact. Later it was discovered that a similar property holds in more general metric spaces: every closed and bounded subset turns out to be compact and conversely, every compact subset is closed and bounded. It turned out that compactness is in fact a covering property: the modern description of compactness via open covers emerged from the work of P. S. Alexandrov and P. S. Urysohn in their famous "Memoire sur les espaces topologiques compacts" [AU29]. Compact spaces in many ways resemble finite sets. For example, the fact that in Hausdorff topological spaces two different points can be separated by disjoint open sets easily generalizes to the fact that in such spaces two disjoint compact sets can also be separated by disjoint open sets. Any finite set is compact in any topology and the fact that compact spaces should be accompanied by some kind of separation axiom comes from the fact that any set with the co-finite topology is compact (but fails to be Hausdorff).However, our favourite "real" objects such as the real line, real plane etc. fail to be compact. Ernst Lindelöf was able to identify the first compactness-like covering property (which was later given his name): the property that from every open cover, one can choose a countable subcover. The Lindelöf theorem, stating that every second countable space is Lindelöf, was proved by him for Euclidean spaces as early as 1903 in [Lin03]. Many facts that held for compact spaces, such as that every closed subspace of a compact space is also compact, remain true in Lindelöf spaces. In metric spaces the Lindelöf property was proved to be equivalent to separability, the existence of a countable basis and the countable chain condition -all of which hold on the real line. But many other properties, such as preservation under products, fail even in the finite case for Lindelöf spaces.Another generalization that is much closer to compactness is the notion of an H-closed space: this is a Hausdorff space in which from every open cover we can choose a finit...

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“…As we pointed out in the beginning, products of compact spaces is compact, and a product of two Lindelöf spaces might not be Lindelöf. Such products might not even be weakly Lindelöf, as the following example from [HJ75] shows:…”

Section: Example ([Ss96]mentioning

confidence: 99%

A Note on Quasi-Lindelof Spaces

Staynova

2012

Preprint

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The quasi-Lindelöf property was first introduced by Arhangelski in [Arc79], as a strengthening of the weakly Lindelöf property. However, unlike Lindelöf and weakly Lindelöf spaces, very little is known about how quasi-Lindelöf spaces behave under the main topological operations, and how the property relates to separation axioms. In the present paper, we look at several properties of quasi-Lindelöf spaces. We consider several examples: a weakly Lindelöf space which is not quasi-Lindelöf, a product of Lindelöf spaces which is not even quasi-Lindelöf, and a quasi-Lindelöf space which is not ccc. At the end, we pose some open questions.

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On the Products of Weakly Lindelof Spaces

Cited by 4 publications

References 0 publications

Weak covering properties and selection principles

Weak covering properties and selection principles

A Comparison of Lindelof-Type Covering Properties of Topological Spaces

A Note on Quasi-Lindelof Spaces

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