scholarly journals L-topological derived internal (resp. Enclosed) relation spaces

Filomat ◽  
2021 ◽  
Vol 35 (8) ◽  
pp. 2497-2516
Author(s):  
Xiu-Yun Wu ◽  
Qi Liu ◽  
Chun-Yan Liao ◽  
Yan-Hui Zhao
Keyword(s):  
Topological Space ◽  
Closure Operator ◽  
Operator Space ◽  
Internal Relation ◽  
Interior Operator

In this paper, notions of L-topological derived internal relation space, L-topological derived interior operator space, L-topological derived enclosed relation space and L-topological derived closure operator space are introduced. It is proved that all of these spaces are categorically isomorphic to L-topological space, L-topological internal relation space and L-topological enclosed relation space.

Fuzzy roughness via ideals

2020 ◽  
Vol 39 (5) ◽  
pp. 6869-6880
Author(s):  
S. H. Alsulami ◽  
Ismail Ibedou ◽  
S. E. Abbas
Keyword(s):  
Fuzzy Set ◽  
Closure Operator ◽  
Approximation Space ◽  
Approximation Spaces ◽  
Fuzzy Connectedness ◽  
Fuzzy Approximation ◽  
Separation Axioms ◽  
Fuzzy Ideal ◽  
Interior Operator

In this paper, we join the notion of fuzzy ideal to the notion of fuzzy approximation space to define the notion of fuzzy ideal approximation spaces. We introduce the fuzzy ideal approximation interior operator int Φ λ and the fuzzy ideal approximation closure operator cl Φ λ , and moreover, we define the fuzzy ideal approximation preinterior operator p int Φ λ and the fuzzy ideal approximation preclosure operator p cl Φ λ with respect to that fuzzy ideal defined on the fuzzy approximation space (X, R) associated with some fuzzy set λ ∈ IX. Also, we define fuzzy separation axioms, fuzzy connectedness and fuzzy compactness in fuzzy approximation spaces and in fuzzy ideal approximation spaces as well, and prove the implications in between.

scholarly journals Fuzzyθ-closure operator on fuzzy topological spaces

1991 ◽  
Vol 14 (2) ◽  
pp. 309-314 ◽  
Author(s):  
M. N. Mukherjee ◽  
S. P. Sinha
Keyword(s):  
Topological Space ◽  
Fuzzy Sets ◽  
Closure Operator ◽  
Topological Spaces ◽  
Separation Axioms ◽  
Fuzzy Topological Space ◽  
Fuzzy Topological Spaces

The paper contains a study of fuzzyθ-closure operator,θ-closures of fuzzy sets in a fuzzy topological space are characterized and some of their properties along with their relation with fuzzyδ-closures are investigated. As applications of these concepts, certain functions as well as some spaces satisfying certain fuzzy separation axioms are characterized in terms of fuzzyθ-closures andδ-closures.

scholarly journals On induced map f# in fuzzy set theory and its applications to fuzzy continuity

2021 ◽  
Author(s):  
Sandeep Kaur ◽  
Nitakshi Goyal
Keyword(s):  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Closure Operator ◽  
Topological Spaces ◽  
Continuous Mappings ◽  
New Vision ◽  
Saturated Sets ◽  
Interior Operator ◽  
Fuzzy Topological Spaces

Abstract In this paper, we introduce # image of a fuzzy set which gives a induced map f # corresponding to any function f : X → Y , where X and Y are crisp sets. With this, we present a new vision of studying fuzzy continuous mappings in fuzzy topological spaces where fuzzy continuity explains the term of closeness in the mathematical models. We also define the concept of fuzzy saturated sets which helps us to prove some new characterizations of fuzzy continuous mappings in terms of interior operator rather than closure operator.

scholarly journals Closure System and Its Semantics

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 198
Author(s):  
Yinbin Lei ◽  
Jun Zhang
Keyword(s):  
Closure Operator ◽  
Spatial Relations ◽  
Accumulation Point ◽  
Boundary Operator ◽  
Topological Spaces ◽  
Closure System ◽  
Derived Set ◽  
Topological Closure ◽  
Interior Operator ◽  
Isolated Point

It is well known that topological spaces are axiomatically characterized by the topological closure operator satisfying the Kuratowski Closure Axioms. Equivalently, they can be axiomatized by other set operators encoding primitive semantics of topology, such as interior operator, exterior operator, boundary operator, or derived-set operator (or dually, co-derived-set operator). It is also known that a topological closure operator (and dually, a topological interior operator) can be weakened into generalized closure (interior) systems. What about boundary operator, exterior operator, and derived-set (and co-derived-set) operator in the weakened systems? Our paper completely answers this question by showing that the above six set operators can all be weakened (from their topological counterparts) in an appropriate way such that their inter-relationships remain essentially the same as in topological systems. Moreover, we show that the semantics of an interior point, an exterior point, a boundary point, an accumulation point, a co-accumulation point, an isolated point, a repelling point, etc. with respect to a given set, can be extended to an arbitrary subset system simply by treating the subset system as a base of a generalized interior system (and hence its dual, a generalized closure system). This allows us to extend topological semantics, namely the characterization of points with respect to an arbitrary set, in terms of both its spatial relations (interior, exterior, or boundary) and its dynamic convergence of any sequence (accumulation, co-accumulation, and isolation), to much weakened systems and hence with wider applicability. Examples from the theory of matroid and of Knowledge/Learning Spaces are used as an illustration.

scholarly journals Strongly Connectedness in Closure Space

2014 ◽  
Vol 9 ◽  
pp. 69-71
Author(s):  
U.D. Tapi ◽  
Bhagyashri A. Deole
Keyword(s):  
Topological Space ◽  
Disjoint Union ◽  
Closure Operator ◽  
Closure Space ◽  
Closed Set ◽  
Power Set ◽  
Strongly Connected

A Čech closure space (X, u) is a set X with Čech closure operator u: P(X) → P(X) where P(X) is a power set of X, which satisfies u𝝓=𝝓, A ⊆uA for every A⊆X, u (A⋃B) = uA⋃uB, for all A, B ⊆ X. Many properties which hold in topological space hold in closure space as well. A topological space X is strongly connected if and only if it is not a disjoint union of countably many but more than one closed set. If X is strongly connected, and Ei‟s are nonempty disjoint closed subsets of X, then X≠ E1∪E2∪. We further extend the concept of strongly connectedness in closure space. The aim of this paper is to introduce and study the concept of strongly connectedness in closure space.

The lattices of families of regular sets in topological spaces

2020 ◽  
Vol 70 (2) ◽  
pp. 477-488
Author(s):  
Emilia Przemska
Keyword(s):  
Closure Operator ◽  
Boolean Algebras ◽  
Topological Spaces ◽  
Closed Sets ◽  
The Family ◽  
Interior Operator ◽  
Regular Sets ◽  
Regular Open ◽  
Complemented Lattice ◽  
Regular Closed

Abstract The question as to the number of sets obtainable from a given subset of a topological space using the operators derived by composing members of the set {b, i, ∨, ∧}, where b, i, ∨ and ∧ denote the closure operator, the interior operator, the binary operators corresponding to union and intersection, respectively, is called the Kuratowski {b, i, ∨, ∧}-problem. This problem has been solved independently by Sherman [21] and, Gardner and Jackson [13], where the resulting 34 plus identity operators were depicted in the Hasse diagram. In this paper we investigate the sets of fixed points of these operators. We show that there are at most 23 such families of subsets. Twelve of them are the topology, the family of all closed subsets plus, well known generalizations of open sets, plus the families of their complements. Each of the other 11 families forms a complete complemented lattice under the operations of join, meet and negation defined according to a uniform procedure. Two of them are the well known Boolean algebras formed by the regular open sets and regular closed sets, any of the others in general need not be a Boolean algebras.

scholarly journals On gω-Open Sets in Grill Topological Spaces

2020 ◽  
pp. 132-143
Author(s):  
Amin Saif ◽  
Mohammed Al-Hawmi ◽  
Basheer Al-Refaei
Keyword(s):  
Closure Operator ◽  
Weak Form ◽  
Topological Spaces ◽  
Cluster Operator ◽  
Open Set ◽  
Interior Operator ◽  
Operator Closure

The propose of this paper is to introduce and investigate a weak form of ω-open set in grill topological spaces. We introduce the notion of -open set as a form stronger than βω-open set and weaker than ω-open set and -open set. By using this form, we study the generalization property, the interior operator, closure operator and θ-cluster operator.

scholarly journals Pointwise k-Pseudo Metric Space

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2505
Author(s):  
Yu Zhong ◽  
Alexander Šostak ◽  
Fu-Gui Shi
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Closure Operator ◽  
Topological Structures ◽  
Neighborhood System ◽  
Interior Operator

In this paper, the concept of a k-(quasi) pseudo metric is generalized to the L-fuzzy case, called a pointwise k-(quasi) pseudo metric, which is considered to be a map d:J(LX)×J(LX)⟶[0,∞) satisfying some conditions. What is more, it is proved that the category of pointwise k-pseudo metric spaces is isomorphic to the category of symmetric pointwise k-remote neighborhood ball spaces. Besides, some L-topological structures induced by a pointwise k-quasi-pseudo metric are obtained, including an L-quasi neighborhood system, an L-topology, an L-closure operator, an L-interior operator, and a pointwise quasi-uniformity.

scholarly journals A NOTE ON ΑIG- CLOSURE AND ΑIG- INTERIOR IN IDEAL TOPOLOGICAL SPACES

2021 ◽  
Vol 12 (4) ◽  
pp. 1276-1279
Author(s):  
D. Vinodhini, Et. al.
Keyword(s):  
Topological Space ◽  
Closure Operator ◽  
Topological Spaces ◽  
Basic Properties

The concepts of αIg- closure, αIg- interior and αIg- boundary of a subset of an ideal topological space (X, t, I) are introduced in this article.  Some of their basic properties are proven. Furthermore, the relationships between these sets are investigated to get the best of them. Also, it is established that αIg- closure is a Kuratowski closure operator on (X, t, I) under certain conditions.

scholarly journals The topology of θω-open sets

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5369-5377 ◽  
Author(s):  
Ghour Al ◽  
Bayan Irshedat
Keyword(s):  
Topological Space ◽  
Closure Operator ◽  
Sufficient Conditions ◽  
Topological Spaces ◽  
Product Theorem ◽  
Closure Operators ◽  
Separation Axiom ◽  
New Class

We define the ??-closure operator as a new topological operator. We show that ??-closure of a subset of a topological space is strictly between its usual closure and its ?-closure. Moreover, we give several sufficient conditions for the equivalence between ??-closure and usual closure operators, and between ??-closure and ?-closure operators. Also, we use the ??-closure operator to introduce ??-open sets as a new class of sets and we prove that this class of sets lies strictly between the class of open sets and the class of ?-open sets. We investigate ??-open sets, in particular, we obtain a product theorem and several mapping theorems. Moreover, we introduce ?-T2 as a new separation axiom by utilizing ?-open sets, we prove that the class of !-T2 is strictly between the class of T2 topological spaces and the class of T1 topological spaces. We study relationship between ?-T2 and ?-regularity. As main results of this paper, we give a characterization of ?-T2 via ??-closure and we give characterizations of ?-regularity via ??-closure and via ??-open sets.

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