More on closed non-vanishing ideals in CB(X)

2020 ◽  
Vol 70 (4) ◽  
pp. 909-916
Author(s):  
Amin Khademi
Keyword(s):  
Supremum Norm ◽  
Topological Properties ◽  
Normed Algebra ◽  
Completely Regular ◽  
Vanishing Ideal ◽  
Algebraic Properties ◽  
Sequential Compactness ◽  
Countable Compactness ◽  
Vanishing Ideals ◽  
The Ideal

AbstractLet X be a completely regular topological space. For each closed non-vanishing ideal H of CB(X), the normed algebra of all bounded continuous scalar-valued mappings on X equipped with pointwise addition and multiplication and the supremum norm, we study its spectrum, denoted by 𝔰𝔭(H). We make a correspondence between algebraic properties of H and topological properties of 𝔰𝔭(H). This continues some previous studies, in which topological properties of 𝔰𝔭(H) such as the Lindelöf property, paracompactness, σ-compactness and countable compactness have been made into correspondence with algebraic properties of H. We study here other compactness properties of 𝔰𝔭(H) such as weak paracompactness, sequential compactness and pseudocompactness. We also study the ideal isomorphisms between two non-vanishing closed ideals of CB(X).

scholarly journals Compactness in Metric Spaces

2016 ◽  
Vol 24 (3) ◽  
pp. 167-172
Author(s):  
Kazuhisa Nakasho ◽  
Keiko Narita ◽  
Yasunari Shidama
Keyword(s):  
Real Line ◽  
Metric Spaces ◽  
Topological Properties ◽  
The Real ◽  
The Third ◽  
Norm Space ◽  
Number Sequences ◽  
Sequential Compactness ◽  
Countable Compactness ◽  
Definition Of

Summary In this article, we mainly formalize in Mizar [2] the equivalence among a few compactness definitions of metric spaces, norm spaces, and the real line. In the first section, we formalized general topological properties of metric spaces. We discussed openness and closedness of subsets in metric spaces in terms of convergence of element sequences. In the second section, we firstly formalize the definition of sequentially compact, and then discuss the equivalence of compactness, countable compactness, sequential compactness, and totally boundedness with completeness in metric spaces. In the third section, we discuss compactness in norm spaces. We formalize the equivalence of compactness and sequential compactness in norm space. In the fourth section, we formalize topological properties of the real line in terms of convergence of real number sequences. In the last section, we formalize the equivalence of compactness and sequential compactness in the real line. These formalizations are based on [20], [5], [17], [14], and [4].

scholarly journals On the gamma spectrum of multiplication gamma acts

2021 ◽  
Vol 48 (2) ◽  
Author(s):  
Mehdi S. Abbas ◽  
◽  
Samer A. Gubeir ◽  
Keyword(s):  
Topological Space ◽  
Gamma Spectrum ◽  
Zariski Topology ◽  
Topological Properties ◽  
Separation Axioms ◽  
Algebraic Properties

In this paper, we introduce the concept of topological gamma acts as a generalization of Zariski topology. Some topological properties of this topology are studied. Various algebraic properties of topological gamma acts have been discussed. We clarify the interplay between this topological space's properties and the algebraic properties of the gamma acts under consideration. Also, the relation between this topological space and (multiplication, cyclic) gamma act was discussed. We also study some separation axioms and the compactness of this topological space.

scholarly journals Sequential completeness of quotient groups

2000 ◽  
Vol 61 (1) ◽  
pp. 129-150 ◽  
Author(s):  
Dikran Dikranjan ◽  
Michael Tkačenko
Keyword(s):  
Abelian Group ◽  
Direct Products ◽  
Topological Groups ◽  
Complete Group ◽  
Sequential Completeness ◽  
Algebraic Properties ◽  
Countable Compactness ◽  
Quotient Groups

We discuss various generalisations of countable compactness for topological groups that are related to completeness. The sequentially complete groups form a class closed with respect to taking direct products and closed subgroups. Surprisingly, the stronger version of sequential completeness called sequential h-completeness (all continuous homomorphic images are sequentially complete) implies pseudocompactness in the presence of good algebraic properties such as nilpotency. We also study quotients of sequentially complete groups and find several classes of sequentially q-complete groups (all quotients are sequentially complete). Finally, we show that the pseudocompact sequentially complete groups are far from being sequentially q-complete in the following sense: every pseudocompact Abelian group is a quotient of a pseudocompact Abelian sequentially complete group.

Zariski Topologies on Graded Ideals

2021 ◽  
Vol 78 (1) ◽  
pp. 215-224
Author(s):  
Malik Bataineh ◽  
Azzh Saad Alshehry ◽  
Rashid Abu-Dawwas
Keyword(s):  
Commutative Ring ◽  
Zariski Topology ◽  
Topological Properties ◽  
Prime Spectrum ◽  
Algebraic Properties

Abstract In this paper, we show there are strong relations between the algebraic properties of a graded commutative ring R and topological properties of open subsets of Zariski topology on the graded prime spectrum of R. We examine some algebraic conditions for open subsets of Zariski topology to become quasi-compact, dense, and irreducible. We also present a characterization for the radical of a graded ideal in R by using topological properties.

scholarly journals Some results involving weak compactness in C(X), C(υX) and C(X)′

1975 ◽  
Vol 19 (3) ◽  
pp. 221-229 ◽  
Author(s):  
I. Tweddle
Keyword(s):  
Dual Space ◽  
Present Note ◽  
Continuous Functions ◽  
Weak Compactness ◽  
Regular Space ◽  
Compact Sets ◽  
Completely Regular ◽  
Countably Compact ◽  
Countable Compactness

The main aim of the present note is to compare C(X) and C(υX), the spaces of real-valued continuous functions on a completely regular space X and its real 1–1 compactification υX, with regard to weak compactness and weak countable compactness. In a sense to be made precise below, it is shown that C(X) and C(υX) have the same absolutely convex weakly countably compact sets. In certain circumstances countable compactness may be replaced by compactness, in which case one obtains a nice representation of the Mackey completion of the dual space of C(X) (Theorems 5, 6, 7).

scholarly journals Ideals of bounded holomorphic functions on simple n-sheeted discs

1991 ◽  
Vol 123 ◽  
pp. 171-201
Author(s):  
Masaru Hara
Keyword(s):  
Riemann Surface ◽  
Banach Algebra ◽  
Positive Constant ◽  
Holomorphic Functions ◽  
Strong Sense ◽  
Supremum Norm ◽  
Corona Problem ◽  
Bounded Holomorphic ◽  
Zero Points ◽  
The Ideal

As usual we denote by H∞(K) the Banach algebra of bounded holomorphic functions on a Riemann surface R equipped with the supremum norm ‖·‖ Consider the ideal I(f1 … fm) of H∞(R) generated by functions f1 …fm in H∞(R). If a function g in H∞(R) belongs to I(f1 … fm or equivalently, if there exist m functions h1 …, hm in H∞(R) withon R, then common zero points of f1, ... fm are also zero points of g in the following strong sense:on R for a positive constant δ > 0. The generalized corona problem asks whether the converse is valid or not. In the case g ≡ 1 on R the problem is referred to simply as the corona problem.

Zariski-like topologies for lattices with applications to modules over associative rings

2019 ◽  
Vol 18 (07) ◽  
pp. 1950131
Author(s):  
Jawad Abuhlail ◽  
Hamza Hroub
Keyword(s):  
Complete Lattice ◽  
Associative Ring ◽  
Sufficient Conditions ◽  
Proper Class ◽  
Topological Properties ◽  
Left Module ◽  
Separation Axioms ◽  
Prime Submodules ◽  
Algebraic Properties ◽  
Associative Rings

We study Zariski-like topologies on a proper class [Formula: see text] of a complete lattice [Formula: see text]. We consider [Formula: see text] with the so-called classical Zariski topology [Formula: see text] and study its topological properties (e.g. the separation axioms, the connectedness, the compactness) and provide sufficient conditions for it to be spectral. We say that [Formula: see text] is [Formula: see text]-top if [Formula: see text] is a topology. We study the interplay between the algebraic properties of an [Formula: see text]-top complete lattice [Formula: see text] and the topological properties of [Formula: see text] Our results are applied to several spectra which are proper classes of [Formula: see text] where [Formula: see text] is a nonzero left module over an arbitrary associative ring [Formula: see text] (e.g. the spectra of prime, coprime, fully prime submodules) of [Formula: see text] as well as to several spectra of the dual complete lattice [Formula: see text] (e.g. the spectra of first, second and fully coprime submodules of [Formula: see text]).

scholarly journals On generalizations of C*-embedding for wallman rings

1978 ◽  
Vol 25 (2) ◽  
pp. 215-229 ◽  
Author(s):  
H. L. Bentley ◽  
B. J. Taylor
Keyword(s):  
Topological Space ◽  
Continuous Functions ◽  
Topological Properties ◽  
Zero Sets ◽  
Algebraic Properties

AbstractBiles (1970) has called a subring A of the ring C(X), of all real valued continuous functions on a topological space X, a Wallman ring on X whenever Z(A), the zero sets of functions belonging to A, forms a normal base on X in the sense of Frink (1964). Previously, we have related algebraic properties of a Wallman ring A to topological properties of the Wallman compactification w(Z(A)) of X determined by the normal base Z(A). Here we introduce two different generalizations of the concept of “a C*-embedded subset” and study relationships between these and topological (respectively, algebraic) properties of w(Z(A)) (respectively, A).

On the second spectrum and the second classical Zariski topology of a module

2015 ◽  
Vol 14 (10) ◽  
pp. 1550150 ◽  
Author(s):  
Seçil Çeken ◽  
Mustafa Alkan
Keyword(s):  
Finite Number ◽  
Associative Ring ◽  
Finite Union ◽  
Zariski Topology ◽  
Topological Properties ◽  
Noetherian Module ◽  
Algebraic Properties ◽  
Second Spectrum

Let R be an associative ring with identity and Specs(M) denote the set of all second submodules of a right R-module M. In this paper, we investigate some interrelations between algebraic properties of a module M and topological properties of the second classical Zariski topology on Specs(M). We prove that a right R-module M has only a finite number of maximal second submodules if and only if Specs(M) is a finite union of irreducible closed subsets. We obtain some interrelations between compactness of the second classical Zariski topology of a module M and finiteness of the set of minimal submodules of M. We give a connection between connectedness of Specs(M) and decomposition of M for a right R-module M. We give several characterizations of a noetherian module M over a ring R such that every right primitive factor of R is artinian for which Specs(M) is connected.

scholarly journals Generalized Differenceλ-Sequence Spaces Defined by Ideal Convergence and the Musielak-Orlicz Function

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Awad A. Bakery
Keyword(s):  
Orlicz Function ◽  
Sequence Spaces ◽  
Complex Numbers ◽  
Infinite Matrix ◽  
Topological Properties ◽  
Normed Spaces ◽  
Ideal Convergence ◽  
Inclusion Relations ◽  
Difference Sequence Spaces ◽  
The Ideal

We introduced the ideal convergence of generalized difference sequence spaces combining an infinite matrix of complex numbers with respect toλ-sequences and the Musielak-Orlicz function overn-normed spaces. We also studied some topological properties and inclusion relations between these spaces.

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