scholarly journals Some New Variants of Relative Regularity via Regularly Closed Sets

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Sehar Shakeel Raina ◽  
A. K. Das
Keyword(s):  
Topological Property ◽  
Topological Properties ◽  
Complete Regularity ◽  
Completely Regular ◽  
Closed Sets ◽  
Relative Property ◽  
Relative Version ◽  
The Absolute

Every topological property can be associated with its relative version in such a way that when smaller space coincides with larger space, then this relative property coincides with the absolute one. This notion of relative topological properties was introduced by Arhangel’skii and Ganedi in 1989. Singal and Arya introduced the concepts of almost regular spaces in 1969 and almost completely regular spaces in 1970. In this paper, we have studied various relative versions of almost regularity, complete regularity, and almost complete regularity. We investigated some of their properties and established relationships of these spaces with each other and with the existing relative properties.

scholarly journals Complete regularity of Ellis semigroups of -actions

2020 ◽  
pp. 1-17
Author(s):  
MARCY BARGE ◽  
JOHANNES KELLENDONK
Keyword(s):  
Complete Regularity ◽  
Completely Regular ◽  
Ellis Semigroup ◽  
Totally Disconnected

Abstract It is shown that the Ellis semigroup of a $\mathbb Z$ -action on a compact totally disconnected space is completely regular if and only if forward proximality coincides with forward asymptoticity and backward proximality coincides with backward asymptoticity. Furthermore, the Ellis semigroup of a $\mathbb Z$ - or $\mathbb R$ -action for which forward proximality and backward proximality are transitive relations is shown to have at most two left minimal ideals. Finally, the notion of near simplicity of the Ellis semigroup is introduced and related to the above.

Separable Topological Space of Hereditary

NUTA Journal ◽  
2020 ◽  
Vol 7 (1-2) ◽  
pp. 68-70
Author(s):  
Raj Narayan Yadav ◽  
Bed Prasad Regmi ◽  
Surendra Raj Pathak
Keyword(s):  
Topological Space ◽  
Topological Property ◽  
Sufficient Condition ◽  
Topological Properties ◽  
Necessary And Sufficient Condition ◽  
Separable Space ◽  
Separable Topological Space ◽  
Necessary And Sufficient

A property of a topological space is termed hereditary ifand only if every subspace of a space with the property also has the property. The purpose of this article is to prove that the topological property of separable space is hereditary. In this paper we determine some topological properties which are hereditary and investigate necessary and sufficient condition functions for sub-spaces to possess properties of sub-spaces which are not in general hereditary.

Descriptive sets

1966 ◽  
Vol 291 (1426) ◽  
pp. 353-367 ◽  
Keyword(s):  
Metric Space ◽  
Regular Space ◽  
Countable Union ◽  
Borel Set ◽  
Borel Sets ◽  
Countable Intersection ◽  
Completely Regular ◽  
Closed Sets ◽  
Open Set

A theory of descriptive Baire sets is developed for an arbitrary completely regular space. It is shown that descriptive Baire sets are Baire sets and that they form a system closed under countable union, countable intersection and intersection with a Baire set. If a descriptive Borel set (Rogers 1965) is a Baire set then it is a descriptive Baire set. If every open set is a countable union of closed sets, the descriptive Baire sets coincide with the descriptive Borel sets. It follows, in particular, that in a metric space a set is descriptive Baire, if, and only if, it is absolutely Borel and Lindelöf.

scholarly journals Infra Soft Compact Spaces and Application to Fixed Point Theorem

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Tareq M. Al-shami
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Topological Properties ◽  
Finite Product ◽  
Finite Intersection ◽  
Compact Spaces ◽  
Closed Sets ◽  
Soft Topology ◽  
Covering Properties

Infra soft topology is one of the recent generalizations of soft topology which is closed under finite intersection. Herein, we contribute to this structure by presenting two kinds of soft covering properties, namely, infra soft compact and infra soft Lindelöf spaces. We describe them using a family of infra soft closed sets and display their main properties. With the assistance of examples, we mention some classical topological properties that are invalid in the frame of infra soft topology and determine under which condition they are valid. We focus on studying the “transmission” of these concepts between infra soft topology and classical infra topology which helps us to discover the behaviours of these concepts in infra soft topology using their counterparts in classical infra topology and vice versa. Among the obtained results, these concepts are closed under infra soft homeomorphisms and finite product of soft spaces. Finally, we introduce the concept of fixed soft points and reveal main characterizations, especially those induced from infra soft compact spaces.

Separating Closed Sets by Continuous Mappings into Developable Spaces

1981 ◽  
Vol 33 (6) ◽  
pp. 1420-1431 ◽  
Author(s):  
Harald Brandenburg
Keyword(s):  
Topological Space ◽  
Double Sequence ◽  
Completely Regular ◽  
Closed Sets ◽  
Continuous Mappings ◽  
Neighbourhood Base ◽  
Image Position ◽  
Metrizable Spaces ◽  
Metrization Theorem ◽  
Developable Spaces

A topological space X is called developable if it has a development, i.e., a sequence of open covers of X such that for each x ∈ X the collection is a neighbourhood base of x, whereThis class of spaces has turned out to be one of the most natural and useful generalizations of metrizable spaces [23]. In [4] it was shown that some well known results in metrization theory have counterparts in the theory of developable spaces (i.e., Urysohn's metrization theorem, the Nagata-Smirnov theorem, and Nagata's “double sequence theorem”). Moreover, in [3] it was pointed out that subspaces of products of developable spaces (i.e., D-completely regular spaces) can be characterized in much the same way as subspaces of products of metrizable spaces (i.e., completely regular T1-spaces).

On the Complete Regularity of Some Category Spaces

1975 ◽  
Vol 17 (5) ◽  
pp. 651-656 ◽  
Author(s):  
W. Eames
Keyword(s):  
Topological Space ◽  
Density Function ◽  
Measure Space ◽  
Baire Property ◽  
Complete Regularity ◽  
Completely Regular ◽  
Upper Density ◽  
Covering Class ◽  
Sequential Covering ◽  
Null Set

A category space is a measure space which is also a topological space, the measure and the topology being related by ‘a set is measurable iff it has the Baire property’ and ‘a set is null iff it is nowhere dense’ [4]. We considered some category spaces in [3]; now we show that if a null set is deleted from the space, then the topology can be taken to be completely regular. The essential part of the construction consists of obtaining a suitable refinement of the original sequential covering class and using the consequent strong upper density function to define the required topology. Then the complete regularity follows much as in [1].

scholarly journals Some Topological Properties ofCbX

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Juan Carlos Ferrando
Keyword(s):  
General Condition ◽  
Regular Space ◽  
Topological Properties ◽  
Compact Sets ◽  
Completely Regular

IfXis a completely regular space, first we characterize those spacesCbXwhose compact sets are metrizable. Then we use this result to provide a general condition forXto ensure the metrizability of compact sets inCbX. Finally, we characterize those spacesCbXthat have aG-basis.

scholarly journals Semi-topological properties

1992 ◽  
Vol 15 (2) ◽  
pp. 267-272
Author(s):  
Bhamini M. P. Nayar ◽  
S. P. Arya
Keyword(s):  
Topological Property ◽  
Topological Properties

A property preserved under a semi-homeomorphism is said to be a semi-topological property. In the present paper we prove the following results: (1) A topological propertyPis semi-topological if and only if the statement(X,𝒯)hasPif and only if(X,F(𝒯))hasP′is true whereF(𝒯)is the finest topology onXhaving the same family of semi-open sets as(X,𝒯), (2) IfPis a topological property being minimalPis semi-topological if and only if for each minimalPspace(X,𝒯),𝒯=F(𝒯).

scholarly journals On lattice-topological properties of general Wallman spaces

1998 ◽  
Vol 21 (1) ◽  
pp. 25-32
Author(s):  
Carmen Vlad
Keyword(s):  
Topological Properties ◽  
Closed Sets ◽  
Finitely Additive Measures ◽  
Additive Measures

LetXbe an arbitrary set andℒa lattice of subsets ofXsuch thatϕ,X∈ℒ.𝒜(ℒ)is the algebra generated byℒandI(ℒ)consists of all zero-one valued finitely additive measures on𝒜(ℒ). Various subsets of andI(ℒ)are considered and certain lattices are investigated as well as the topology of closed sets generated by them. The lattices are investigated for normality, regularity, repleteness and completeness. The topologies are similarly discussed for various properties such asT2and Lindelöf.

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).

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