Complete Hausdorffness and Complete Regularity on Supra Topological Spaces

The supra topological topic is of great importance in preserving some topological properties under conditions weaker than topology and constructing a suitable framework to describe many real-life problems. Herein, we introduce the version of complete Hausdorffness and complete regularity on supra topological spaces and discuss their fundamental properties. We show the relationships between them with the help of examples. In general, we study them in terms of hereditary and topological properties and prove that they are closed under the finite product space. One of the issues we are interested in is showing the easiness and diversity of constructing examples that satisfy supra T i spaces compared with their counterparts on general topology.

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On Soft Separation Axioms and Their Applications on Decision-Making Problem

Mathematical Problems in Engineering ◽

10.1155/2021/8876978 ◽

2021 ◽

Vol 2021 ◽

pp. 1-12

Author(s):

T. M. Al-shami

Keyword(s):

Decision Making ◽

Topological Spaces ◽

Topological Properties ◽

Finite Product ◽

Separation Axioms ◽

Decision Making Problem ◽

Weak Structure

In this work, we introduce new types of soft separation axioms called p t -soft α regular and p t -soft α T i -spaces i = 0,1,2,3,4 using partial belong and total nonbelong relations between ordinary points and soft α -open sets. These soft separation axioms enable us to initiate new families of soft spaces and then obtain new interesting properties. We provide several examples to elucidate the relationships between them as well as their relationships with e -soft T i , soft α T i , and t t -soft α T i -spaces. Also, we determine the conditions under which they are equivalent and link them with their counterparts on topological spaces. Furthermore, we prove that p t -soft α T i -spaces i = 0,1,2,3,4 are additive and topological properties and demonstrate that p t -soft α T i -spaces i = 0,1,2 are preserved under finite product of soft spaces. Finally, we discuss an application of optimal choices using the idea of p t -soft T i -spaces i = 0,1,2 on the content of soft weak structure. We provide an algorithm of this application with an example showing how this algorithm is carried out. In fact, this study represents the first investigation of real applications of soft separation axioms.

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Beta- Star- Continuity and Beta- Star- Contra- Continuity

International Journal of Pure Mathematics ◽

10.46300/91019.2020.7.6 ◽

2021 ◽

Vol 7 ◽

pp. 43-66

Author(s):

Raja Mohammad Latif

Keyword(s):

Continuous Functions ◽

General Topology ◽

Topological Spaces ◽

New Class ◽

Fundamental Properties ◽

Open Set ◽

Class Of Functions

In 2014 Mubarki, Al-Rshudi, and Al- Juhani introduced and studied the notion of a set in general topology called β*-open set and investigated its fundamental properties and studied the relationships between β*-open set and other topological sets including β*-continuity in topological spaces. We introduce and investigate several properties and characterizations of a new class of functions between topological spaces called β*- open, β*- closed, β*- continuous and β*- irresolute functions in topological spaces. We also introduce slightly β*- continuous, totally β*- continuous and almost β*- continuous functions between topological spaces and establish several characterizations of these new forms of functions. Furthermore, we also introduce and investigate certain ramifications of contra continuous and allied functions, namely, contra β*- continuous, and almost contra β*-continuous functions along with their several properties, characterizations and natural relationships. Moreover, we introduce new types of closed graphs by using β*- open sets and investigate its properties and characterizations in topological spaces.

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Limit Points and Separation Axioms with Respect to Supra Semi-open Sets

European Journal of Pure and Applied Mathematics ◽

10.29020/nybg.ejpam.v13i3.3743 ◽

2020 ◽

Vol 13 (3) ◽

pp. 427-443 ◽

Cited By ~ 1

Author(s):

Tareq M. AL-shami ◽

E. A. Abo-Tabl ◽

Baravan Assad ◽

Mohamed Arahet

Keyword(s):

Real Life ◽

Research Area ◽

Topological Spaces ◽

Difference Property ◽

Separation Axioms ◽

Life Problems ◽

New Concepts ◽

Limit Points ◽

The Difference

Sometimes we need to minimize the conditions of topology for different reasons such as obtaining more convenient structures to describe some real-life problems, or constructing some counterexamples whom show the interrelations between certain topological concepts, or preserving some properties under fewer conditions of those on topology. To contribute this research area, in this paper, we establish some new concepts on supra topological spaces using supra semi-open sets and give some characterizations of them. First, we introduce a concept of supra semi limit points of a set and study main properties, in particular, on the spaces that possess the difference property. Second, we define and investigate new separation axioms, namely supra semi Ti-spaces (i = 0, 1, 2, 3, 4) and give complete descriptions for each one of them. We provide some examples to show the relationships between them as well as with STi-space.

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On product of spaces of quasicomponents

Filomat ◽

10.2298/fil1720307m ◽

2017 ◽

Vol 31 (20) ◽

pp. 6307-6311

Author(s):

Gjorgji Markoski ◽

Abdulla Buklla

Keyword(s):

Product Space ◽

Continuous Functions ◽

Topological Spaces ◽

Topological Properties ◽

Locally Compact

We use a characterization of quasicomponents by continuous functions to obtain the well known theorem which states that product of quasicomponents Qx,Qy of topological spaces X,Y, respectively, gives quasicomponent in the product space X x Y. If spaces X,Y are locally-compact, paracompact and Haussdorf, then we prove that the space of quasicomponents of the product Q(XxY) is homeomorphic with the product space Q(X) x Q(Y), so these two spaces have the same topological properties.

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β* - and β* - Connectedness

WSEAS TRANSACTIONS ON MATHEMATICS ◽

10.37394/23206.2020.19.16 ◽

2020 ◽

Vol 19 ◽

Keyword(s):

Compact Space ◽

General Topology ◽

Topological Spaces ◽

Connected Space ◽

Fundamental Properties ◽

Open Set ◽

New Concepts

In 2014 Mubarki, Al-Rshudi, and Al-Juhani introduced and studied the notion of a set in general topology called β * - open sets and investigated its fundamental properties and studied the relationships between β * - open set and other topological sets including β * - continuity in topological spaces. The objective of this paper is to introduce the new concepts called β * - compact space, countably β * - compact space, β * - Lindelof space, almost β * - compact space, mildly β * - compact space and β * - connected space in general topology and investigate several properties and characterizations of these new concepts in topological spaces.

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Infra -α- Compact and Infra -α- Connected Spaces

International Journal of Pure Mathematics ◽

10.46300/91019.2021.8.6 ◽

2021 ◽

Vol 8 ◽

pp. 41-57

Author(s):

Raja Mohammad Latif

Keyword(s):

Compact Space ◽

General Topology ◽

Topological Spaces ◽

Connected Space ◽

Fundamental Properties ◽

Open Set ◽

New Concepts ◽

The Relationship ◽

Connected Spaces

In 2016 Hakeem A. Othman and Md. Hanif Page introduced a new notion of set in general topology called an infra -α- open set and investigated its fundamental properties and studied the relationship between infra -α- open set and other topological sets. The objective of this paper is to introduce the new concepts called infra -α- compact space, countably infra -α- compact space, infra -α- Lindelof space, almost infra -α- compact space, mildly infra -α- compact space and infra -α- connected space in general topology and investigate several properties and characterizations of these new concepts in topological spaces.

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Topological properties of spaces ordered by preferences

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171299220170 ◽

1999 ◽

Vol 22 (1) ◽

pp. 17-27 ◽

Cited By ~ 5

Author(s):

J. C. R. Alcantud

Keyword(s):

Order Topology ◽

Topological Spaces ◽

Topological Properties ◽

Binary Relations ◽

Complete Regularity ◽

Relevant Class

In this paper, we analyze the main topological properties of a relevant class of topologies associated with spaces ordered by preferences (asymmetric, negatively transitive binary relations). This class consists of certain continuous topologies which include the order topology. The concept of saturated identification is introduced in order to provide a natural proof of the fact that all these spaces possess topological properties analogous to those of linearly ordered topological spaces, inter alia monotone and hereditary normality, and complete regularity.

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Partial soft separation axioms and soft compact spaces

Filomat ◽

10.2298/fil1813755e ◽

2018 ◽

Vol 32 (13) ◽

pp. 4755-4771 ◽

Cited By ~ 13

Author(s):

M.E. El-Shafei ◽

M. Abo-Elhamayel ◽

T.M. Al-Shami

Keyword(s):

Topological Space ◽

General Topology ◽

Regular Space ◽

Topological Spaces ◽

Finite Product ◽

Compact Spaces ◽

Separation Axioms ◽

Soft Topological Space

The main aim of the present paper is to define new soft separation axioms which lead us, first, to generalize existing comparable properties via general topology, second, to eliminate restrictions on the shape of soft open sets on soft regular spaces which given in [22], and third, to obtain a relationship between soft Hausdorff and new soft regular spaces similar to those exists via general topology. To this end, we define partial belong and total non belong relations, and investigate many properties related to these two relations. We then introduce new soft separation axioms, namely p-soft Ti-spaces (i = 0,1,2,3,4), depending on a total non belong relation, and study their features in detail. With the help of examples, we illustrate the relationships among these soft separation axioms and point out that p-soft Ti-spaces are stronger than soft Ti-spaces, for i = 0,1,4. Also, we define a p-soft regular space, which is weaker than a soft regular space and verify that a p-soft regular condition is sufficient for the equivalent among p-soft Ti-spaces, for i = 0,1,2. Furthermore, we prove the equivalent among finite p-soft Ti-spaces, for i = 1,2,3 and derive that a finite product of p-soft Ti-spaces is p-soft Ti, for i = 0,1,2,3,4. In the last section, we show the relationships which associate some p-soft Ti-spaces with soft compactness, and in particular, we conclude under what conditions a soft subset of a p-soft T2-space is soft compact and prove that every soft compact p-soft T2-space is soft T3-space. Finally, we illuminate that some findings obtained in general topology are not true concerning soft topological spaces which among of them a finite soft topological space need not be soft compact.

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