On 𝑆-closed spaces

Joseph, James E.; Kwack, Myung H.

doi:10.1090/s0002-9939-1980-0577771-4

Cited by 71 publications

(13 citation statements)

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“…The θ-semiclosure [11] of A, denoted by s Cl θ (A), is defined to be the set of all x ∈ X such that A ∩ Cl(U ) = ∅ for every semiopen set U containing x. A subset A is called θ-semiclosed [11] if and only if A = s Cl θ (A). The complement of a θ-semiclosed set is called a θ-semiopen set [11].…”

Section: Preliminariesmentioning

confidence: 99%

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ON UPPER AND LOWER ALMOST CONTRA-$\mathcal{I}$-CONTINUOUS MULTIFUNCTIONS

Arivazhai¹,

Rajesh²,

Shanthi³

2017

Int. J. of Pure and Appl. Math.

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Section: Preliminariesmentioning

confidence: 99%

“…A subset A is called θ-semiclosed [11] if and only if A = s Cl θ (A). The complement of a θ-semiclosed set is called a θ-semiopen set [11]. The family of all regular open (resp.…”

Section: Preliminariesmentioning

confidence: 99%

ON UPPER AND LOWER ALMOST CONTRA-$\mathcal{I}$-CONTINUOUS MULTIFUNCTIONS

Arivazhai¹,

Rajesh²,

Shanthi³

2017

Int. J. of Pure and Appl. Math.

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“…The semi-interior of a set A is the union of all semi-open sets contained in A and is denoted by sIntA. A subset A of a topological space (X, τ ) is said to be θ-open [23] (resp., θ-semi-open [13], semi-θ-open [6]) set if for each x ∈ A, there is an open (resp., semi-open, semi-open) set U such that x ∈ U ⊆ Cl(U ) ⊆ A (resp., x ∈ U ⊆ Cl(U ) ⊆ A, x ∈ U ⊆ sCl(U ) ⊆ A). For more properties of semi-θ-open sets (see [24]) also.…”

Section: ) Set If a ⊆ Intcl(a) (Resp A ⊆ Intclint(a) A ⊆ Clint(a) mentioning

confidence: 99%

Ss-Open sets and Ss-Continuous Functions

Khalaf

Majeed²,

Jamil³

2013

International Journal of Advanced Mathematical Sciences

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In this paper we introduce and study the concept of S s -open sets .also, a study new class of functions called S S continuous functions, the relationships between S s -continuity and other types of continuity are investigated. Keywords: S s -open set, S s -continuous function, semi-open set, semi-continuous functions. IntroductionIn 1963, Levine [16], introduced the concept of semi-open set and semi continuity and gave several properties about these functions. Njastad [18] introduced the concepts of α-sets and Abd-El-Monsef et al [1] defined β-open sets and β-continuous functions. Khalaf and Ameen in [14] PreliminariesIn this section, we recall the following definitions and results:Lemma 2.2 Let A be a subset of a space X, then the following properties hold. Lemma 2.7 The following properties hold: semi-regular [11], if for each x ∈ X and each H ∈ SO(X) containing x, there exists1. If a space X is semi-regular, then each SO(X) = SθO(X). If a space X is semi-regular, then sCl( Proof. It is clear that each semi-θ-open is semi-open. If X is semi-regular space and if G is a non-empty semi-open set in X, the by Definition 2.5, there exists a semi-open set U such that x ∈ U ⊆ sCl(U ) ⊆ G, this implies that G is semi-θ-open. Therefore, SO(X) = SθO(X). Part (2). Follows from part (1). Definition 2.8 A space X is locally indiscrete [9], if every open set is closed. Lemma 2.9 [9] A space X is locally indiscrete if and only if every semi open set in X is closed. Definition 2.10 [19] A function f : X → Y is said to be strongly θ-semi-continuous at a point x ∈ X, if for each open set V containing f (x), there exists a semi-open set U containingProof. Follows from the fact that in a semi-T 1 space, every singleton set is semi-closed (Lemma 2.6).Remark 3.

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“…A point x ∈ X is said to be a θ-semi-cluster point [25] denoted by δO(X, x) (resp.GO(X, x),GC(X, x), πGO (X, x), πGC(X, x), RO(X, x), RC(X, x), SO(X, x), C(X, x)). open, regular closed) sets of X is denoted by δO(X) (respGO(X),GC(X), πGO(X), πGC(X), SO(X), βO(X), PO(X), RO(X), RC(X)).…”

Section: The Finite Union Of Regular Open Set Is Said To Be π-Openmentioning

confidence: 99%