Displaying similar documents to “Countable compactness and p -limits”

Some results on quasi-t-dual Baer modules

Rachid Tribak, Yahya Talebi, Mehrab Hosseinpour (2023)

Commentationes Mathematicae Universitatis Carolinae

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Let R be a ring and let M be an R -module with S = End R ( M ) . Consider the preradical Z ¯ for the category of right R -modules Mod- R introduced by Y. Talebi and N. Vanaja in 2002 and defined by Z ¯ ( M ) = ⋂ { U ≤ M : M / U is small in its injective hull } . The module M is called quasi-t-dual Baer if ∑ ϕ ∈ ℑ ϕ ( Z ¯ 2 ( M ) ) is a direct summand of M for every two-sided ideal ℑ of S , where Z ¯ 2 ( M ) = Z ¯ ( Z ¯ ( M ) ) . In this paper, we show that M is quasi-t-dual Baer if and only if Z ¯ 2 ( M ) is a direct summand of M and Z ¯ 2 ( M ) is a quasi-dual Baer module. It is also shown that any direct...

Order boundedness and weak compactness of the set of quasi-measure extensions of a quasi-measure

Zbigniew Lipecki (2015)

Commentationes Mathematicae Universitatis Carolinae

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Let 𝔐 and ℜ be algebras of subsets of a set Ω with 𝔐 ⊂ ℜ , and denote by E ( μ ) the set of all quasi-measure extensions of a given quasi-measure μ on 𝔐 to ℜ . We give some criteria for order boundedness of E ( μ ) in b a ( ℜ ) , in the general case as well as for atomic μ . Order boundedness implies weak compactness of E ( μ ) . We show that the converse implication holds under some assumptions on 𝔐 , ℜ and μ or μ alone, but not in general.

A note on star Lindelöf, first countable and normal spaces

Wei-Feng Xuan (2017)

Mathematica Bohemica

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A topological space X is said to be star Lindelöf if for any open cover 𝒰 of X there is a Lindelöf subspace A ⊂ X such that St ( A , 𝒰 ) = X . The “extent” e ( X ) of X is the supremum of the cardinalities of closed discrete subsets of X . We prove that under V = L every star Lindelöf, first countable and normal space must have countable extent. We also obtain an example under MA + ¬ CH , which shows that a star Lindelöf, first countable and normal space may not have countable extent.

Boundedness of sublinear operators in Triebel-Lizorkin spaces via atoms

Liguang Liu, Dachun Yang (2009)

Studia Mathematica

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Let s ∈ ℝ, p ∈ (0,1] and q ∈ [p,∞). It is proved that a sublinear operator T uniquely extends to a bounded sublinear operator from the Triebel-Lizorkin space Ḟ p , q s ( ℝ ⁿ ) to a quasi-Banach space ℬ if and only if sup | | T ( a ) | | ℬ : a is an infinitely differentiable (p,q,s)-atom of Ḟ p , q s ( ℝ ⁿ ) < ∞, where the (p,q,s)-atom of Ḟ p , q s ( ℝ ⁿ ) is as defined by Han, Paluszyński and Weiss.

An observation on spaces with a zeroset diagonal

Wei-Feng Xuan (2020)

Mathematica Bohemica

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We say that a space X has the discrete countable chain condition (DCCC for short) if every discrete family of nonempty open subsets of X is countable. A space X has a zeroset diagonal if there is a continuous mapping f : X 2 → [ 0 , 1 ] with Δ X = f - 1 ( 0 ) , where Δ X = { ( x , x ) : x ∈ X } . In this paper, we prove that every first countable DCCC space with a zeroset diagonal has cardinality at most 𝔠 .

Nearstandardness on a finite set

Lyantse V.

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AbstractLet T be a finite set for which card T is a natural nonstandard number. The linear space ℂ T of complex-valued functions on T is nonstandard. For the analysis on ℂ T we need a concept of nearstandardness in this space. A version how to introduce such a concept is proposed. Some elementary examples are given. CONTENTSIntroduction.................................................................................................................50. Preliminary notes....................................................................................................7 0.1....

On subcompactness and countable subcompactness of metrizable spaces in ZF

Kyriakos Keremedis (2022)

Commentationes Mathematicae Universitatis Carolinae

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We show in ZF that: (i) Every subcompact metrizable space is completely metrizable, and every completely metrizable space is countably subcompact. (ii) A metrizable space 𝐗 = ( X , T ) is countably compact if and only if it is countably subcompact relative to T . (iii) For every metrizable space 𝐗 = ( X , T ) , the following are equivalent: (a) 𝐗 is compact; (b) for every open filter ℱ of 𝐗 , ⋂ { F ¯ : F ∈ ℱ } ≠ ∅ ; (c) 𝐗 is subcompact relative to T . We also show: (iv) The negation of each of the statements, (a) every countably subcompact...

Spaces with star countable extent

A. D. Rojas-Sánchez, Angel Tamariz-Mascarúa (2016)

Commentationes Mathematicae Universitatis Carolinae

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For a topological property P , we say that a space X is star P if for every open cover 𝒰 of the space X there exists A ⊂ X such that s t ( A , 𝒰 ) = X . We consider space with star countable extent establishing the relations between the star countable extent property and the properties star Lindelöf and feebly Lindelöf. We describe some classes of spaces in which the star countable extent property is equivalent to either the Lindelöf property or separability. An example is given of a Tychonoff star Lindelöf...

More reflections on compactness

Lúcia R. Junqueira, Franklin D. Tall (2003)

Fundamenta Mathematicae

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We consider the question of when X M = X , where X M is the elementary submodel topology on X ∩ M, especially in the case when X M is compact.

A nice subclass of functionally countable spaces

Vladimir Vladimirovich Tkachuk (2018)

Commentationes Mathematicae Universitatis Carolinae

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A space X is functionally countable if f ( X ) is countable for any continuous function f : X → ℝ . We will call a space X exponentially separable if for any countable family ℱ of closed subsets of X , there exists a countable set A ⊂ X such that A ∩ ⋂ 𝒢 ≠ ∅ whenever 𝒢 ⊂ ℱ and ⋂ 𝒢 ≠ ∅ . Every exponentially separable space is functionally countable; we will show that for some nice classes of spaces exponential separability coincides with functional countability. We will also establish that the class of exponentially separable...

Quasi-completeness on the Spaces of Holomorphic Germs

Roberto Luiz Soraggi (1981)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Sia E uno spazio D ⁢ F riflessivo e sia K un compatto di E . Si dimostra che lo spazio dei germi olomorfi su K , con la topologia naturale, è un limite induttivo regolare e quasi completo purché lo spazio dei germi olomorfi all'origine sia un limite induttivo regolare.