Cardinal inequalities for topological spaces involving the weak Lindelof number

Bell, Murray G.; Ginsburg, John; Woods, R. Grant

doi:10.2140/pjm.1978.79.37

Cited by 48 publications

(53 citation statements)

References 6 publications

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“…In literature there are several generalizations of the notion of Lindelof space [2] and these are studied separately for different reasons and purposes In 1959 Frolik [3] introduced the notion of weaklyLindel0f spaces that, aPterward, was studied by several authors: Comfort, Hindman and Negrepointis [4] in 1969, Hager [5] in 1969, Ulmer [6] in 1972, Woods [7] in 1976, Bell, Ginsburg and Woods [8] in 1978 About this topic in 1982 Balasubramanian [9] introduced and studied the notion of nearly-Lindelof spaces that is between Lindel0f and weakly-Lindelof spaces In 1984 Willard and Dissanayake 10] gave the notion of almost k-Lindel0f space, that for k N0 we call almost-Lindelof, and that is between the nearly-Lindelof and weakly-Lindelof spaces. To be complete, it is useful to recall some recent papers of Pareek 11 which are an almost survey of all main generalizations of Lindelof spaces…”

Section: Introductionmentioning

confidence: 99%

Some counterexamples and properties on generalizations of Lindelöf spaces

Cammaroto

Santoro

1995

International Journal of Mathematics and Mathematical Sciences

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Section: Introductionmentioning

confidence: 99%

Some counterexamples and properties on generalizations of Lindelöf spaces

Cammaroto

Santoro

1995

International Journal of Mathematics and Mathematical Sciences

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“…In 1993 Alas proved that |X| 2 χ(X)wL c (X) holds for the class of Urysohn spaces, thereby generalizing both (5) and (6). Prior to this result, Bella and Cammaroto had obtained the inequality |X| 2 χ(X)aL(X) for Urysohn spaces.…”

Section: Generalizations and Variations Of |X| 2 χ (X)l(x)mentioning

confidence: 99%

“…The inequality |X| 2 χ(X)wL(X) for normal spaces is due to Bell et al [6]; this was the first variation of Arhangel'skiȋ's inequality to use the cardinal function wL. At about the same time, Arhangel'skiȋ [3] proved the inequality |X| 2 χ(X)wL c (X) for regular spaces.…”

Section: Generalizations and Variations Of |X| 2 χ (X)l(x)mentioning

confidence: 99%

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Arhangel'skiĭ's solution to Alexandroff's problem: A survey

Hodel

2006

Topology and its Applications

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In 1969, Arhangel'skiȋ proved that |X| 2 χ(X)L(X) for every Hausdorff space X. This beautiful inequality solved a nearly fifty-year old question raised by Alexandroff and Urysohn. In this paper we survey a wide range of generalizations and variations of Arhangel'skiȋ's inequality. We also discuss open problems and an important legacy of the theorem: the emergence of the closure method as a fundamental unifying device in cardinal functions.

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“…Thus the weak Lindelöf property is somewhere between being a covering property and a chain condition. Woods [16] used the weak Lindelöf property to characterize the C * -embedded subsets of the Stone-Cech compactification of the integers under CH and Bell Ginsburgh and Woods [5] exploited it in their elegant generalization of Arhangel'skii's theorem on the cardinality of compact first-countable spaces.…”

Section: Introductionmentioning

confidence: 99%

Infinite games and chain conditions

Spadaro

2016

Fund. Math.

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Abstract. We apply the theory of infinite two-person games to two well-known problems in topology: Suslin's Problem and Arhangel'skii's problem on G δ covers of compact spaces. More specifically, we prove results of which the following two are special cases: 1) every linearly ordered topological space satisfying the game-theoretic version of the countable chain condition is separable and 2) in every compact space satisfying the game-theoretic version of the weak Lindelöf property, every cover by G δ sets has a continuum-sized subcollection whose union is G δ -dense. IntroductionInfinite games have been exploited in recent years to give partial answers to various important problems in General Topology, including van Douwen's D-space problem (see [1]), Arhangel'skii's problem on the cardinality of Lindelöf spaces with points G δ (see [14], [2]) and Bell, Ginsburg and Woods's problem on the cardinality of weakly Lindelöf first-countable regular spaces (see [6], [3], [4]). We use them to give partial ZFC answers to Suslin's Problem and Arhangel'skii's question of whether in every compact space, a cover by G δ sets has a continuum-sized subfamily with a G δ -dense union.It was already known to Cantor that the real line is the unique complete dense linear order without endpoints which is separable. In the first issue of Fundamenta Mathematicae, Suslin asked whether separability could be replaced with the countable chain condition in this result. Any counterexample to this assertion came to be known as a Suslin Line. The problem turned out to be independent of the usual axioms of ZFC: under MA ω 1 there are no Suslin Lines, and thus the answer to Suslin's question is yes. However, Suslin Lines can be found in certain models of set theory, for example under V = L. Various mathematicians wondered whether there was a natural strengthening of the ccc which implied a positive answer to Suslin's Problem. For example, Knaster proved that every ordered continuum with the Knaster property is separable and Shapirovskii proved that every compact space with countable tightness and Shanin's condition is separable (see [15]).2000 Mathematics Subject Classification. Primary: 54A25, 91A44; Secondary: 54F05, 54G10.

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Cardinal inequalities for topological spaces involving the weak Lindelof number

Cited by 48 publications

References 6 publications

Some counterexamples and properties on generalizations of Lindelöf spaces

Some counterexamples and properties on generalizations of Lindelöf spaces

Arhangel'skiĭ's solution to Alexandroff's problem: A survey

Infinite games and chain conditions

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