Semi-Open Sets and Semi-Continuity in Topological Spaces

Levine, Norman

doi:10.2307/2312781

Cited by 549 publications

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“…In 1963, Levine [5] introduced semi-open sets. The term preopen subset was introduced by Mashhour, Abd El-Monsef and ElDeeb [10] but the concept had appeared much earlier.…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

On faint continuity

McCluskey

Reilly

2015

Appl.Gen.Topol.

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Recently the class of strongly faintly α-continuous functions between topological spaces has been defined and studied in some detail. We consider this class of functions from the perspective of change(s) of topology. In particular, we conclude that each member of this class of functions belongs the usual class of continuous functions between topological spaces when the domain and codomain of the function in question have been retopologized appropriately. Some consequences of this fact are considered in this paper.2010 MSC: 06A06; 54H10.

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“…In 1963, Levine [5] introduced semi-open sets. The term preopen subset was introduced by Mashhour, Abd El-Monsef and ElDeeb [10] but the concept had appeared much earlier.…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

On faint continuity

McCluskey

Reilly

2015

Appl.Gen.Topol.

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“…Specifically, if τ X and τ Y are (ordinary) topologies on X and Y , then (τ X , τ Y )-continuity is classical topological continuity. Furthermore, (sτ X , τ Y )-continuity is the semi-continuity of [10], (pτ X , τ Y )-continuity is precontinuity in the sense of [11], (sτ X , sτ Y )-continuous functions are defined to be irresolute in [2], while (pτ X , pτ Y )-continuous functions are called preirresolute in [16]. So these are all examples of the use of the change of generalized topology technique to reduce a particular property of functions between ordinary topological spaces to generalized continuity.…”

Section: Change Of Generalized Topologymentioning

confidence: 99%

On g-α-irresolute functions

Bai

Zuo

2010

Acta Math Hung

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“…A set A in a space X is called preopen [11] (respectively, semiopen [6] and α-open [13] The intersection of all preclosed sets containing a subset A is called the preclosure [2] of A and is denoted by Pcl(A). The preinterior of A is the union of all preopen sets of X contained in A.…”

Section: Preliminaries Throughout This Paper (X τ) and (Y σ )mentioning

confidence: 99%

On almost precontinuous functions

Jafari

Noiri

1998

International Journal of Mathematics and Mathematical Sciences

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Abstract. Nasef and Noiri (1997) introduced and investigated the class of almost precontinuous functions. In this paper, we further investigate some properties of these functions.Keywords and phrases. Preopen, preclosed, preboundary, strongly compact, almost precontinuous, strongly prenormal.2000 Mathematics Subject Classification. Primary 54C10; Secondary 54C08. Singal and Singal [24] introduced the notion of almost continuity. Feeble continuity was introduced by Maheshwari et al. [8]. As a generalization of almost continuity and feeble continuity, Maheshwari et al. [7] introduced the notion of almost feeble continuity. Nasef and Noiri [12] introduced a new class of functions called almost precontinuous functions. They showed that almost precontinuity is a generalization of each of almost feeble continuity and almost α-continuity [17]. Introduction.The purpose of this paper is to investigate some more properties of almost precontinuous functions. It turns out that almost precontinuity is stronger than almost weak continuity introduced by Jankovic [5]. The intersection of all preclosed sets containing a subset A is called the preclosure [2] of A and is denoted by Pcl(A). The preinterior of A is the union of all preopen sets of X contained in A. The family of all preopen sets of X will be denoted by PO(X). For a point x of X, we put PO(X, PreliminariesIf f is almost continuous at every point of X, then it is called almost continuous. If f is almost precontinuous at every point of X, then it is called almost precontinuous. Example 2.10. Let X = {a, b, c, d} and τ = {X, ∅, {b}, {c}, {a, b}, {b, c}, {a, b, c}, {b, c, d}} We have the following result. Now suppose that X is submaximal. Let x be a point in X and V any regular open set containing f (x). Since X is submaximal, every preopen set of X is open [22, Theorem 4]. If we set Ᏺ = PO(X, x), then Ᏺ will be a filter base which p-converges to x. So there exists U in Ᏺ such that f (U) ⊂ V . This completes the proof.The following corollary is suggested by the referee. If f is w.α.c. and g is a.p.c., then the x) by Lemma 3.9 and O ∩ E = ∅. Therefore, we obtain x ∉ Pcl(E). This shows that E is preclosed in X. Corollary 3.11 (Popa [19]). Let f ,g : X → Y be functions and Y Hausdorff. If f is continuous and g is precontinuous, then the setTherefore, we obtain (x 1 ,x 2 ) ∈ Pcl(E). This shows that E is preclosed in X 1 × X 2 . Corollary 3.13. If Y is Hausdorff and f : X → Y is an a.p.c. function, then the setProof. By setting X = X 1 = X 2 and g = f in Theorem 3.12, the result follows. We introduce the following concept.

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Semi-Open Sets and Semi-Continuity in Topological Spaces

Cited by 549 publications

References 0 publications

On faint continuity

On faint continuity

On g-α-irresolute functions

On almost precontinuous functions

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