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.

SYMMETRY GEOMETRY BY PAIRINGS

2018 ◽  
Vol 106 (03) ◽  
pp. 342-360 ◽  
Author(s):  
G. CHIASELOTTI ◽  
T. GENTILE ◽  
F. INFUSINO
Keyword(s):  
Closure Operator ◽  
Local Symmetry ◽  
Global Symmetry ◽  
Symmetry Relation ◽  
Basic Tool ◽  
Mathematical Structures ◽  
Finite Case ◽  
Set Systems ◽  
Power Set ◽  
Abstract Simplicial Complex

In this paper, we introduce asymmetry geometryfor all those mathematical structures which can be characterized by means of a generalization (which we call pairing) of a finite rectangular table. In more detail, let$\unicode[STIX]{x1D6FA}$be a given set. Apairing$\mathfrak{P}$on$\unicode[STIX]{x1D6FA}$is a triple$\mathfrak{P}:=(U,F,\unicode[STIX]{x1D6EC})$, where$U$and$\unicode[STIX]{x1D6EC}$are nonempty sets and$F:U\times \unicode[STIX]{x1D6FA}\rightarrow \unicode[STIX]{x1D6EC}$is a map having domain$U\times \unicode[STIX]{x1D6FA}$and codomain$\unicode[STIX]{x1D6EC}$. Through this notion, we introduce a local symmetry relation on$U$and a global symmetry relation on the power set${\mathcal{P}}(\unicode[STIX]{x1D6FA})$. Based on these two relations, we establish the basic properties of our symmetry geometry induced by$\mathfrak{P}$. The basic tool of our study is a closure operator$M_{\mathfrak{P}}$, by means of which (in the finite case) we can represent any closure operator. We relate the study of such a closure operator to several types of others set operators and set systems which refine the notion of an abstract simplicial complex.

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 Fine Fuzzy sp Closed Sets in Fine Fuzzy Topological Spaces

2020 ◽  
Vol 9 (3) ◽  
pp. 1306-1313
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Inverse Image ◽  
Continuous Functions ◽  
Topological Spaces ◽  
The Novel ◽  
Closed Set ◽  
Closed Sets ◽  
Fuzzy Topological Space ◽  
Fuzzy Topological Spaces

The main view of this article is the extended version of the fine topological space to the novel kind of space say fine fuzzy topological space which is developed by the notion called collection of quasi coincident of fuzzy sets. In this connection, fine fuzzy closed sets are introduced and studied some features on it. Further, the relationship between fine fuzzy closed sets with certain types of fine fuzzy closed sets are investigated and their converses need not be true are elucidated with necessary examples. Fine fuzzy continuous function is defined as the inverse image of fine fuzzy closed set is fine fuzzy closed and its interrelations with other types of fine fuzzy continuous functions are obtained. The reverse implication need not be true is proven with examples. Finally, the applications of fine fuzzy continuous function are explained by using the composition.

scholarly journals Relations of Pre Generalized Regular Weakly Locally Closed Sets in Topological Spaces

2021 ◽  
Vol 12 (3) ◽  
pp. 3444-3454
Author(s):  
Vijayakumari T Et.al
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Closed Set ◽  
Closed Sets ◽  
Open Set ◽  
Continuous Maps ◽  
Locally Closed Sets

In this paper pgrw-locally closed set, pgrw-locally closed*-set and pgrw-locally closed**-set are introduced. A subset A of a topological space (X,t) is called pgrw-locally closed (pgrw-lc) if A=GÇF where G is a pgrw-open set and F is a pgrw-closed set in (X,t). A subset A of a topological space (X,t) is a pgrw-lc* set if there exist a pgrw-open set G and a closed set F in X such that A= GÇF. A subset A of a topological space (X,t) is a pgrw-lc**-set if there exists an open set G and a pgrw-closed set F such that A=GÇF. The results regarding pgrw-locally closed sets, pgrw-locally closed* sets, pgrw-locally closed** sets, pgrw-lc-continuous maps and pgrw-lc-irresolute maps and some of the properties of these sets and their relation with other lc-sets are established.

scholarly journals On the Weak Order of Coxeter Groups

2019 ◽  
Vol 71 (2) ◽  
pp. 299-336 ◽  
Author(s):  
Matthew Dyer
Keyword(s):  
Complete Lattice ◽  
Coxeter Groups ◽  
Closure Operator ◽  
Root Systems ◽  
Bruhat Order ◽  
Weak Order ◽  
Maximal Subgroups ◽  
Rank Zero ◽  
Power Set ◽  
Rank One

AbstractThis paper provides some evidence for conjectural relations between extensions of (right) weak order on Coxeter groups, closure operators on root systems, and Bruhat order. The conjecture focused upon here refines an earlier question as to whether the set of initial sections of reflection orders, ordered by inclusion, forms a complete lattice. Meet and join in weak order are described in terms of a suitable closure operator. Galois connections are defined from the power set of $W$ to itself, under which maximal subgroups of certain groupoids correspond to certain complete meet subsemilattices of weak order. An analogue of weak order for standard parabolic subsets of any rank of the root system is defined, reducing to the usual weak order in rank zero, and having some analogous properties in rank one (and conjecturally in general).

Large families of incomparable A-isols

1983 ◽  
Vol 48 (2) ◽  
pp. 250-252 ◽  
Author(s):  
William S. Heck
Keyword(s):  
Natural Number ◽  
Natural Partial Order ◽  
List Type ◽  
Type Number ◽  
Closed Set ◽  
Strict Inclusion ◽  
Power Set ◽  
Large Families ◽  
Areas Of Interest ◽  
The Continuum

The set Λ of isols was extensively studied by Dekker and Myhill in [1]. Subsequently, Nerode [3] developed the theory of Λ(A), the set of isols relative to some recursively closed set of functions A.One of the main areas of interest of [1] was the natural partial order ≤ on Λ. In this paper we will examine some of the properties of ≤A on Λ(A). We use the following notations: ∣A∣ is the cardinality of the set A, ⊃ denotes strict inclusion, (a) is the power set of the set a, c is the cardinality of the continuum, and ω = {0, 1, 2, …}. The terms A-isol, A-immune, A-r.e., A-incomparable, etc. all refer to the usual meaning of these words, only taken in the context of the recursively closed set A. ReqA(a) is the A-r.e.t. of which a is a representative. By identifying a finite natural number with the A-r.e.t. consisting of sets of a given finite cardinality we see that ω ⊆ Λ(A); Λ(A) is said to be nontrivial iff ω ⊃ Λ(A). The three results proven in this paper are:Theorem 1. If Λ(A) is nontrivial, then ∣Λ(A)∣ = c.Theorem 2. If∣A∣ < c, then Λ(A) is nontrivial.Theorem 3. If ∣A∣ < c and ∣⊿∣ < c and ⊿ ⊆ Λ(A) − ω, then there is aΓ ⊆ Λ(A) − ω such that:(a) ∣Γ∣ = c.(b) Every member of Γ is A-incomparable with every member of Δ.(c) Any two distinct members of Γ are A-incomparable.

scholarly journals A VIEW TO POWER SET: A TOPOLOGICAL DISCUSSION

2019 ◽  
pp. 1-2
Author(s):  
aripex Amuly
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Topological Properties ◽  
Power Set ◽  
The Given

A Power Set is not only a container of all family of subsets of a set and the set itself,but ,in topology,it is also a generator of all topologies on the defined set. So, there is a topological existence of power set, being the strongest topology ever defined on a set,there are some properties of it's topological existence.In this paper, such properties are being proved and concluded. The following theorems stated are on the basis of the topological properties and separated axioms,which by satisfying, moves to a conclusion that,not only a power set is just a topology on the given defined set,but also it can be considered as a “Universal Topology”or a “Universal Topological Space”,that is the container of all topological spaces. This paper gives a general understanding about what a power set is,topologically and gives us a new perceptive from a “power set”to a “topological power space”.

scholarly journals Extensions of closure spaces

2003 ◽  
Vol 4 (2) ◽  
pp. 223
Author(s):  
D. Deses ◽  
A. De Groot-Van der Voorde ◽  
E. Lowen-Colebunders
Keyword(s):  
Closure Operator ◽  
Finite Additivity ◽  
Closure Space ◽  
Closure Spaces ◽  
The One ◽  
Separation Conditions

<p>A closure space X is a set endowed with a closure operator P(X) → P(X), satisfying the usual topological axioms, except finite additivity. A T<sub>1</sub> closure extension Y of a closure space X induces a structure ϒ on X satisfying the smallness axioms introduced by H. Herrlich [?], except the one on finite unions of collections. We'll use the word seminearness for a smallness structure of this type, i.e. satisfying the conditions (S1),(S2),(S3) and (S5) from [?]. In this paper we show that every T<sub>1</sub> seminearness structure ϒ on X can in fact be induced by a T<sub>1</sub> closure extension. This result is quite different from its topological counterpart which was treated by S.A. Naimpally and J.H.M. Whitfield in [?]. Also in the topological setting the existence of (strict) extensions satisfying higher separation conditions such as T<sub>2</sub> and T<sub>3</sub> has been completely characterized by means of concreteness, separatedness and regularity [?]. In the closure setting these conditions will appear to be too weak to ensure the existence of suitable (strict) extensions. In this paper we introduce stronger alternatives in order to present internal characterizations of the existence of (strict) T<sub>2</sub> or strict regular closure extensions.</p>

scholarly journals On r -Generalized Fuzzy ℓ -Closed Sets: Properties and Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
I. M. Taha
Keyword(s):  
Topological Space ◽  
Closed Set ◽  
Closed Sets ◽  
Separation Axioms ◽  
Connected Sets ◽  
Fuzzy Ideal

In the present study, we introduce and characterize the class of r -generalized fuzzy ℓ -closed sets in a fuzzy ideal topological space X , τ , ℓ in Šostak sense. Also, we show that r -generalized fuzzy closed set by Kim and Park (2002) ⟹ r -generalized fuzzy ℓ -closed set, but the converse need not be true. Moreover, if we take ℓ = ℓ 0 , the r -generalized fuzzy ℓ -closed set and r -generalized fuzzy closed set are equivalent. After that, we define fuzzy upper (lower) generalized ℓ -continuous multifunctions, and some properties of these multifunctions along with their mutual relationships are studied with the help of examples. Finally, some separation axioms of r -generalized fuzzy ℓ -closed sets are introduced and studied. Also, the notion of r -fuzzy G ∗ -connected sets is defined and studied with help of r -generalized fuzzy ℓ -closed sets.

scholarly journals Soft idealization of a decomposition theorem

Filomat ◽  
2016 ◽  
Vol 30 (3) ◽  
pp. 741-751 ◽  
Author(s):  
Yunus Yumak ◽  
Aynur Kaymakcı
Keyword(s):  
Topological Space ◽  
Decomposition Theorem ◽  
Topological Spaces ◽  
Closed Set ◽  
Soft Topological Space ◽  
Regular Closed

Firstly, we define a new set called soft regular-?-closed set in soft ideal topological spaces and obtain some properties of it. Secondly, to obtain a decomposition of continuity in these spaces, we introduce two notions of soft Hayashi-Samuels space and soft A?-set. Finally, we give a decomposition of continuity in a domain Hayashi-Samuels space and in a range soft topological space.

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