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Franklin D Tall - One of the best experts on this subject based on the ideXlab platform.
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FIRST Countable SPACES WITH CALIBER ℵ1 MAY OR MAY NOT BE SEPARABLE
Set-Theoretic Topology, 2014Co-Authors: Franklin D TallAbstract:Publisher Summary This chapter discusses the first Countable vspaces with caliber 81. A space has caliber 81 if every unCountable collection of open sets includes an unCountable subcollection with nonempty intersection. This property lies strictly between the Countable Chain Condition and separability. It is proved that the continuum hypothesis implies first Countable Hausdorff spaces with caliber 81 are separable. A simple proof can be obtained by noting that first Countable Hausdorff spaces satisfying the Countable Chain Condition have cardinality ≤280 [J, 2.16] and that spaces of cardinality 81 with caliber 81 are separable.
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Locally compact perfectly normal spaces may all be paracompact
Fundamenta Mathematicae, 2010Co-Authors: Paul B Larson, Franklin D TallAbstract:We work towards establishing that if it is consistent that there is a supercompact cardinal then it is consistent that every locally compact perfectly normal space is paracompact. At a crucial step we use some still unpublished results announced by Todorcevic. Modulo this and the large cardinal, this answers a question of S. Watson. Modulo these same unpublished results, we also show that if it is consistent that there is a supercompact cardinal, it is consistent that every locally compact space with a hereditarily normal square is metrizable. We also solve a problem raised by the second author, proving it consistent with ZFC that every rst Countable hereditarily normal Countable Chain Condition space is hereditarily separable.
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compact spaces elementary submodels and the Countable Chain Condition
Annals of Pure and Applied Logic, 2006Co-Authors: Lucia R Junqueira, Paul B Larson, Franklin D TallAbstract:Given a space 〈X,J〉 in an elementary submodel M of H(θ), define XM to be X∩M with the topology generated by {U∩M:U∈J∩M}. It is established, using anti-large-cardinals assumptions, that if XM is compact and its regular open algebra is isomorphic to that of a continuous image of some power of the two-point discrete space, then X=XM. Assuming CH + SCH (the Singular Cardinals Hypothesis) in addition, the result holds for any compact XM satisfying the Countable Chain Condition.
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compact spaces elementary submodels and the Countable Chain Condition ii
Topology and its Applications, 2006Co-Authors: Franklin D TallAbstract:Abstract Given a space 〈 X , T 〉 in an elementary submodel of H ( θ ) , define X M to be X ∩ M with the topology generated by { U ∩ M : U ∈ T ∩ M } . It is established that if X M is compact and satisfies the Countable Chain Condition, while X is not scattered and has cardinality less than the first inaccessible cardinal, then X = X M . If the character of X M is a member of M, then “inaccessible” may be replaced by “1-extendible”.
D W Mcintyre - One of the best experts on this subject based on the ideXlab platform.
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a regular Countable Chain Condition space without compact caliber ω1 ω
Annals of the New York Academy of Sciences, 1993Co-Authors: D W McintyreAbstract:. A topological space has the Countable Chain Condition (CCC) if every disjoint collection of nonempty open sets is Countable. It has compact-caliber (ω1, ω) if, for every family {Uα: α∈ω1) of nonempty open sets, there is a compact set K such that K ∩ Uα |Mn O for infinitely many α∈ω1. It has been previously shown that CCC implies compact-caliber (ω1, ω) for first Countable regular spaces. An example is constructed to show that CCC does not imply compact-caliber (ω1, ω) for arbitrary regular spaces. The method of construction is to refine the usual topology on the set of real numbers, and take the Pixley-Roy space over this refinement.
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A Regular Countable Chain Condition Space without Compact‐caliber (ω1, ω)
Annals of the New York Academy of Sciences, 1993Co-Authors: D W McintyreAbstract:. A topological space has the Countable Chain Condition (CCC) if every disjoint collection of nonempty open sets is Countable. It has compact-caliber (ω1, ω) if, for every family {Uα: α∈ω1) of nonempty open sets, there is a compact set K such that K ∩ Uα |Mn O for infinitely many α∈ω1. It has been previously shown that CCC implies compact-caliber (ω1, ω) for first Countable regular spaces. An example is constructed to show that CCC does not imply compact-caliber (ω1, ω) for arbitrary regular spaces. The method of construction is to refine the usual topology on the set of real numbers, and take the Pixley-Roy space over this refinement.
Wei-feng Xuan - One of the best experts on this subject based on the ideXlab platform.
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More on cellular-Lindelöf spaces
Topology and its Applications, 2019Co-Authors: Wei-feng Xuan, Yan-kui SongAbstract:Abstract The class of cellular-Lindelof spaces was introduced by A. Bella and S. Spadaro (2017) [5] . Recall that a topological space X is cellular-Lindelof if for every family U of pairwise disjoint non-empty open sets of X there is a Lindelof subspace L ⊂ X such that U ∩ L ≠ ∅ , for every U ∈ U . Cellular-Lindelof spaces generalize both Lindelof spaces and spaces with the Countable Chain Condition. In this paper, we first discuss some basic properties of cellular-Lindelof spaces such as the behavior with respect to products and subspaces. We also establish cardinal inequalities for cellular-Lindelof quasitopological groups by using Erdos-Rado's theorem. Finally, we introduce and study the class of cellular-compact (cellular-σ-compact) spaces. In particular, we prove that every cellular-σ-compact Hausdorff space having either a rank 2-diagonal or a regular G δ -diagonal has cardinality at most c , which partially answers Question 8 and Question 9 of S. Spadaro and A. Bella (2018) [6] . Some new questions are also posed.
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Symmetric g-functions and cardinal inequalities
Topology and its Applications, 2017Co-Authors: Wei-feng XuanAbstract:Abstract In this paper, we prove that the cardinality of a space X with a symmetric g -function such that ∩ { g 2 ( n , x ) : n ∈ ω } = { x } is at most c if X satisfies one of the following Conditions: (1) X has Countable Chain Condition; (2) X is star Countable (even star σ -compact); (3) X is DCCC (defined below) and normal space. We also prove that if X is a DCCC space with a symmetric g -function such that ∩ { g 3 ( n , x ) : n ∈ ω } = { x } then the cardinality of X is at most c . Finally, we make some observations on Moore spaces.
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A note on spaces with a rank 2-diagonal
Bulletin of The Australian Mathematical Society, 2014Co-Authors: Wei-feng XuanAbstract:We prove that if a space \(X\) with a rank 2-diagonal either has the Countable Chain Condition or is star Countable then the cardinality of \(X\) is at most \(c\). DOI: 10.1017/S0004972713001184
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A note on spaces with a rank \(3\)-diagonal
Bulletin of The Australian Mathematical Society, 2014Co-Authors: Wei-feng XuanAbstract:We prove that if \(X\) is a space satisfying the discrete Countable Chain Condition with a rank 3-diagonal then the cardinality of \(X\) is at most \(\mathfrak c\). DOI:- 10.1017/S0004972714000318
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A NOTE ON SPACES WITH RANK 2-DIAGONAL
Bulletin of The Australian Mathematical Society, 2014Co-Authors: Wei-feng XuanAbstract:We prove that if a space $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}X$ with a rank 2-diagonal either has the Countable Chain Condition or is star Countable then the cardinality of $X$ is at most $\mathfrak{c}$ .
Teruyuki Yorioka - One of the best experts on this subject based on the ideXlab platform.
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Todorcevic orderings as examples of ccc forcings without adding random reals
Commentationes Mathematicae Universitatis Carolinae, 2015Co-Authors: Teruyuki YoriokaAbstract:In [Two examples of Borel partially ordered sets with the Countable Chain Condition, Proc. Amer. Math. Soc. 112 (1991), no. 4, 1125–1128], Todorcevic introduced a ccc forcing which is Borel definable in a separable metric space. In [On Todorcevic orderings, Fund. Math., to appear], Balcar, Pazak and Thummel applied it to more general topological spaces and called such forcings Todorcevic orderings. There they analyze Todorcevic orderings quite deeply. A significant remark is that Thummel solved the problem of Horn and Tarski by use of Todorcevic ordering [The problem of Horn and Tarski, Proc. Amer. Math. Soc. 142 (2014), no. 6, 1997–2000]. This paper supplements the analysis of Todorcevic orderings due to Balcar, Pazak and Thummel in [On Todorcevic orderings, Fund. Math., to appear]. More precisely, it is proved that Todorcevic orderings add no random reals whenever they have the Countable Chain Condition.
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Uniformizing ladder system colorings and the rectangle refining property
Proceedings of the American Mathematical Society, 2010Co-Authors: Teruyuki YoriokaAbstract:We investigate forcing notions with the rectangle refining property, which is stronger than the Countable Chain Condition, and fragments of Martin's Axiom for such forcing notions. We prove that it is consistent that every forcing notion with the rectangle refining property has precaliber H 1 but MA N1 for forcing notions with the rectangle refining property fails.
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A non-implication between fragments of Martin's Axiom related to a property which comes from Aronszajn trees
Annals of Pure and Applied Logic, 2009Co-Authors: Teruyuki YoriokaAbstract:Abstract We introduce a property of forcing notions, called the anti- R 1 , ℵ 1 , which comes from Aronszajn trees. This property canonically defines a new Chain Condition stronger than the Countable Chain Condition, which is called the property R 1 , ℵ 1 . In this paper, we investigate the property R 1 , ℵ 1 . For example, we show that a forcing notion with the property R 1 , ℵ 1 does not add random reals. We prove that it is consistent that every forcing notion with the property R 1 , ℵ 1 has precaliber ℵ 1 and MA ℵ 1 for forcing notions with the property R 1 , ℵ 1 fails. This negatively answers a part of one of the classical problems about implications between fragments of MA ℵ 1 .
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Rudin's Dowker space in the extension with a Suslin tree
Fundamenta Mathematicae, 2008Co-Authors: Teruyuki YoriokaAbstract:We introduce a generalization of a Dowker space constructed from a Suslin tree by Mary Ellen Rudin, and the rectangle rening property for forcing notions, which modies the one for partitions due to Paul B. Larson and Stevo Todor cevi c and is stronger than the Countable Chain Condition. It is proved that Martin's Axiom for forcing no- tions with the rectangle rening property implies that every generalized Rudin space constructed from Aronszajn trees is non-Dowker, and that the same can be forced with a Suslin tree. Moreover, we consider generalized Rudin spaces constructed with some types of non-Aronszajn !1-trees under the Proper Forcing Axiom.
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forcings with the Countable Chain Condition and the covering number of the marczewski ideal
Archive for Mathematical Logic, 2003Co-Authors: Teruyuki YoriokaAbstract:We prove that the covering number of the Marczewski ideal is equal to ℵ1 in the extension with the iteration of Hechler forcing.
Thomas Jech - One of the best experts on this subject based on the ideXlab platform.
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Weak distributivity, a problem of Von Neumann and the mystery of measurability
The Bulletin of Symbolic Logic, 2006Co-Authors: Bohuslav Balcar, Thomas JechAbstract:�DedicatedtoDorothyMaharamStone This article investigates the weak distributivity of Booleano-algebras satisfying the Countable Chain Condition. It addresses primarily the question when such algebras carry ao-additive measure. We use as a starting point the problem of John von Neumann stated in 1937 in the Scottish Book. He asked if the Countable Chain Condition and weak distributivity are sufficient for the existence of such a measure. Subsequent research has shown that the problem has two aspects: one set theoretic and one combinatorial. Recent results provide a complete solution of both the set theoretic and the combinatorial problems. We shall survey the history of von Neumann’s Problem and outline the solution of the set theoretic problem. The technique that we describe owes much to the early work of Dorothy Maharam to whom we dedicate this article. §1. CompleteBooleanalgebrasandweakdistributivity. ABooleanalgebra
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Some Applications of Forcing
Set Theory, 1997Co-Authors: Thomas JechAbstract:The real line is, up to isomorphism, the unique linearly ordered set that is dense, unbounded, complete, and separable. In 1920 Suslin raised the question as to whether “ separable “ can be replaced by a weaker Condition: Each collection of disjoint open intervals is at most Countable (the Countable Chain Condition, c.c.c).
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Generalized iteration of forcing
Transactions of the American Mathematical Society, 1991Co-Authors: M. Groszek, Thomas JechAbstract:Generalized iteration extends the usual notion of iterated forcing from iterating along an ordinal to iterating along any partially ordered set. We consider a class of forcings called perfect tree forcing. The class includes Axiom A forcings with a finite splitting property, such as Cohen, Laver, Mathias, Miller, Prikry-Silver, and Sacks forcings. If S is a perfect tree forcing, there is a decomposition @ * M such that @ is countably closed, M has the Countable Chain Condition, and @ * M adds a p-generic set. Theorem. The mixed-support generalized iteration of perfect tree forcing decompositions along any well-founded partial order preserves cow . Theorem. If ZFC is consistent, so is ZFC + 2' is arbitrarily large + whenever Y is a perfect tree forcing and 0 is a collection of co, dense subsets of _, there is a !i-generic filter on Y.
