scholarly journals Theory of g-Tg-Derived and g-Tg-Coderived Operators: Definitions, Essential Properties, Iterations, and Ranks

2020 ◽  
Author(s):  
Mohammad Irshad KHODABOCUS ◽  
Noor-Ul-Hacq SOOKIA
Keyword(s):  
Topological Space ◽  
Closure Operators ◽  
Closed Sets ◽  
Essential Properties ◽  
Generalized Topological Space ◽  
Delta G ◽  
Transfinite Recursion

In a generalized topological space Tg = (Ω, Tg) (Tg-space), the g-topology Tg : P (Ω) −→ P (Ω) can be characterized in the generalized sense by specifying the generalized open, generalized closed sets (g-Tg-open, g-Tg-closed sets), generalized interior, generalized closure operators g-Intg, g-Clg : P (Ω) −→ P (Ω) (g-Tg-interior, g-Tg-closure operators), or generalized derived, generalized coderived operators g-Derg, g-Codg : P (Ω) −→ P (Ω) (g-Tg-derived, g-Tg-coderived operators), respectively. For very many Tg-spaces, the δth-iterates g-Derg(δ), g-Codg(δ) : P (Ω) −→ P (Ω) of g-Derg, g-Codg : P (Ω) −→ P (Ω), respectively, defined by transfinite recursion on the class of successor ordinals are also themselves g-Tg-derived, g-Tg-coderived operators for new g-topologies in the generalized sense on Ω. Thus, the use of novel definitions of g-Tg-derived, g-Tg-coderived operators g-Derg, g-Codg : P (Ω) −→ P (Ω), respectively, based on a very clever construction, together with their δth-iterates g-Tg-operators g-Derg(δ), g-Codg(δ) : P (Ω) −→ P (Ω), defined by transfinite recursion on the class of successor ordinals, will give rise to novel generalized g-topologies on Ω. The present authors have been actively engaged in the study of g-Tg-operators in Tg-spaces. The study of the essential properties and the commutativity of novel definitions of g-Tg-interior and g-Tg-closure operators g-Intg, g-Clg : P (Ω) −→ P (Ω), respectively, in Tg has formed the first part, and the study of the essential properties and sets of consistent, independent axioms of novel definitions of g-Tg-exterior and g-Tg-frontier operators g-Extg, g-Frg : P (Ω) −→ P (Ω), respectively, has formed the second part. In this work, which forms the last part on the theory of g-Tg-operators in Tg-spaces, the present authors propose to present novel definitions and the study of the essential properties of g-Tg-derived and g-Tg-coderived operators g-Derg, g-Codg : P (Ω) −→ P (Ω), respectively, and their δth-iterates, and the notions of g-Tg-open and g-Tg-closed sets of ranks δ in Tg-spaces.

scholarly journals Theory of g-Tg-Interior and g-Tg-Closure Operators: Definitions, Essential Properties, and Commutativity

2019 ◽  
Author(s):  
Mohammad Irshad KHODABOCUS ◽  
Noor-Ul-Hacq SOOKIA
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Closure Operators ◽  
New Class ◽  
Essential Properties ◽  
Generalized Topological Space ◽  
Outstanding Result

In a generalized topological space Tg = (Ω, Tg), ordinary interior and ordinary closure operators intg, clg : P (Ω) −→ P (Ω), respectively, are defined in terms of ordinary sets. In order to let these operators be as general and unified a manner as possible, and so to prove as many generalized forms of some of the most important theorems in generalized topological spaces as possible, thereby attaining desirable and interesting results, the present au- thors have defined the notions of generalized interior and generalized closure operators g-Intg, g-Clg : P (Ω) −→ P (Ω), respectively, in terms of a new class of generalized sets which they studied earlier and studied their essen- tial properties and commutativity. The outstanding result to which the study has led to is: g-Intg : P (Ω) −→ P (Ω) is finer (or, larger, stronger) than intg : P (Ω) −→ P (Ω) and g-Clg : P (Ω) −→ P (Ω) is coarser (or, smal ler, weaker) than clg : P (Ω) −→ P (Ω). The elements supporting this fact are reported therein as a source of inspiration for more generalized operations.

scholarly journals Theory of g-Tg-Exterior and g-Tg-Frontier Operators: Definitions, Essential Properties and, Consistent, Independent Axioms

2020 ◽  
Author(s):  
Mohammad Irshad Khodabocus ◽  
Noor-Ul-Hacq Soolia
Keyword(s):  
Topological Space ◽  
Recent Work ◽  
Closure Operators ◽  
Closure Operations ◽  
Essential Properties ◽  
Generalized Topological Space

In a generalized topological space Tg = (Ω, Tg), generalized interior and generalized closure operators g-Intg, g-Clg : P (Ω) −→ P (Ω), respectively, are merely two of a number of generalized primitive operators which may be employed to topologize the underlying set Ω in the generalized sense. Generalized exterior and generalized frontier operators g-Extg, g-Frg : P (Ω) −→ P (Ω), respectively, are other generalized primitive operators by means of which characterizations of generalized operations under g-Intg, g-Clg : P (Ω) −→ P (Ω) can be given without even realizing generalized interior and generalized closure operations first in order to topologize Ω in the generalized sense. In a recent work, the present authors have defined novel types of generalized interior and generalized closure operators g-Intg, g-Clg : P (Ω) −→ P (Ω), respectively, in Tg and studied their essential properties and commutativity. In this work, they propose to present novel definitions of generalized exterior and generalized frontier operators g-Extg, g-Frg : P (Ω) −→ P (Ω), respectively, a set of consistent, independent axioms after studying their essential properties, and established further characterizations of generalized operations under g-Intg, g-Clg : P (Ω) −→ P (Ω) in Tg.

scholarly journals On generalized γµ-closed sets and related continuity

2021 ◽  
Vol 13 (2) ◽  
pp. 483-493
Author(s):  
Ritu Sen
Keyword(s):  
Topological Space ◽  
Main Interest ◽  
Closed Sets ◽  
Separation Axioms ◽  
Preservation Theorems ◽  
Generalized Topological Space ◽  
New Type ◽  
Weak Separation

Abstract In this paper our main interest is to introduce a new type of generalized open sets defined in terms of an operation on a generalized topological space. We have studied some properties of this newly defined sets. As an application, we have introduced some weak separation axioms and discussed some of their properties. Finally, we have studied some preservation theorems in terms of some irresolute functions.

Δμ-sets and ∇μ-sets in generalized topological spaces

2017 ◽  
Vol 24 (3) ◽  
pp. 403-407
Author(s):  
Pon Jeyanthi ◽  
Periadurai Nalayini ◽  
Takashi Noiri
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Closed Sets ◽  
Generalized Topological Space

AbstractIn this paper, we introduce and study some properties of the sets, namely {\Delta_{\mu}}-sets, {\nabla_{\mu}}-sets and {\Delta_{\mu}^{*}}-closed sets in a generalized topological space.

scholarly journals A Class of Separation Axioms in Generalized Topology

2016 ◽  
Vol 4 (2) ◽  
pp. 151-159
Author(s):  
D Anabalan ◽  
Santhi C
Keyword(s):  
Topological Space ◽  
Generalized Topology ◽  
Regular Space ◽  
Topological Spaces ◽  
Closed Sets ◽  
New Class ◽  
Separation Axioms ◽  
Generalized Topological Space

The purpose of this paper is to introduce and study some new class of definitions like µ-point closure and gµ –regular space concerning generalized topological space. We obtain some characterizations and several properties of such definitions. This paper takes some investigations on generalized topological spaces with gµ –closed sets and gµ–closed sets.

scholarly journals New Soft Structure: Infra Soft Topological Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Tareq M. Al-shami
Keyword(s):  
Topological Space ◽  
Soft Set ◽  
Topological Spaces ◽  
Closure Operators ◽  
Relative Topology ◽  
Closed Sets ◽  
Soft Topology ◽  
Soft Limit ◽  
Basic Concepts ◽  
Soft Topological Space

It is always convenient to find the weakest conditions that preserve some topologically inspired properties. To this end, we introduce the concept of an infra soft topology which is a collection of subsets that extend the concept of soft topology by dispensing with the postulate that the collection is closed under arbitrary unions. We study the basic concepts of infra soft topological spaces such as infra soft open and infra soft closed sets, infra soft interior and infra soft closure operators, and infra soft limit and infra soft boundary points of a soft set. We reveal the main properties of these concepts with the help of some elucidative examples. Then, we present some methods to generate infra soft topologies such as infra soft neighbourhood systems, basis of infra soft topology, and infra soft relative topology. We also investigate how we initiate an infra soft topology from crisp infra topologies. In the end, we explore the concept of continuity between infra soft topological spaces and determine the conditions under which the continuity is preserved between infra soft topological space and its parametric infra topological spaces.

scholarly journals e#-open and *e-open sets via -open sets

2021 ◽  
Vol 12 (2) ◽  
pp. 2125-2128
Author(s):  
V. Amsaveni, Et. al.
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Closure Operators ◽  
Closed Sets

The notion of -open sets in a topological space was studied by Velicko.  Usha Parmeshwari et.al. and Indira et.al. introduced the concepts of b# and *b open sets respectively. Following this Ekici et. al. studied the notions of e-open and e-closed sets by mixing the closure, interior, -interior and -closure operators.  In this paper some new sets namely e#-open and *e-open sets are defined and their relationship with other similar concetps in topological spaces will be investigated.

scholarly journals On (μ1, μ2, μ3)-Weakly Generalized Closed Sets

2020 ◽  
Vol 13 (4) ◽  
pp. 977-986
Author(s):  
Breix Michael Agua ◽  
Rolando N. Paluga
Keyword(s):  
Topological Space ◽  
Closed Set ◽  
Closed Sets ◽  
Special Cases ◽  
Generalized Topological Space

This paper defines a new generalization of closed sets in a tri-generalized topological space called (μ1, μ2, μ3)-weakly generalized closed set (or briefly (μ1,μ2, μ3)-wg closed set) which is defined as follows: A subset A of X is (μ1, μ2, μ3)-weakly generalized closed set if clμ1(intμ2(A)) ⊆U whenever A ⊆ U and U is μ3-open in X. At least fifteen defined closed sets found in literature are considered special cases of (μ1, μ2, μ3)-weakly generalized closed set under some conditions. Furthermore, some properties of (μ1, μ2, μ3)-weakly generalized closed sets are obtained.

scholarly journals Λµ- sets and V µ- sets in generalized topological spaces

2017 ◽  
Vol 35 (1) ◽  
pp. 33
Author(s):  
P Jeyanthi ◽  
P Nalayini ◽  
T Noiri
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Closed Sets ◽  
Generalized Topological Space

In this paper, we introduce and study some properties of the new sets namely * Λµ- sets , * V µ- sets, * λµ- closed sets, * λµ- open sets in generalized topological space.

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