On some classes of nearly open sets

Njåstad, Olav

doi:10.2140/pjm.1965.15.961

Cited by 718 publications

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“…We denote the interior and the closure of a set A by Int(A) and Cl(A), respectively. A subset A of X is said to be -open [5] if A Int(Cl(Int(A))). The complement of an -open set is said to be -closed.…”

Section: Preliminariesmentioning

confidence: 99%

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α-ℽ-convergence, α-ℽ- accumulation and α-ℽ- compactness

Alias¹,

İbrahim²

2017

Communications Faculty of Science University of Ankara Series A1Mathematics and Statistics

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Section: Preliminariesmentioning

confidence: 99%

“…The productivity and fruitfulness of this notion of compactness motivated mathematicians to generalize this notion. Njastad [5] introduced a new class of generalized open sets in a topological space called -open sets. The concept of operation was initiated by Ibrahim [6].…”

Section: Introductionmentioning

confidence: 99%

α-ℽ-convergence, α-ℽ- accumulation and α-ℽ- compactness

Alias¹,

İbrahim²

2017

Communications Faculty of Science University of Ankara Series A1Mathematics and Statistics

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“…PO (X, τ ); SO (X, τ ); SPO (X, τ )]. The family α(X, τ ) is a topology on X larger than τ , in general [12]. The complement of an α-open [resp.…”

Section: Preliminariesmentioning

confidence: 99%

On some classes of sets in extremally disconnected spaces

Duszyński

2011

Demonstratio Mathematica

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Abstract. In the present paper several characterizations of the classical notion of extremally disconnected spaces are obtained. A few relationships for finite products of extremally disconnected spaces are also studied. IntroductionExtremally disconnected topological spaces were introduced by Gillman and Jerison [6]. It is well-known that each extremally disconnected (e.d., in brief) compact and Hausdorff space is called a Stonean space (for instance, the Stone-Čech compactification βN is a compact, Hausdorff and e.d. space). E.d. spaces play an important role in the theory of Boolean algebras and in some branches of functional analysis. There is a duality between Stonean spaces and the category of complete Boolean algebras. The importance of e.d. space becomes clearer in a study of absolute topological spaces. The purpose of this paper is to give several characterizations of e. [8,9,16,18]), where no separation axioms are assumed. Nevertheless, it is worth to observe that some finite spaces that we make use of as examples, may fulfil weaker forms of separation. For example: the space in Example 1 is semi-T 2 [10] (but not even T 1 ). PreliminariesA topological space on which no separation axiom is assumed will be denoted by (X, τ ). For a subset S of (X, τ ) the closure of S and the interior of S (in (X, τ )) are denoted by cl (S) and int (S) respectively. A subset S of The family of all α-open [resp. preopen; semiopen; semi-preopen] subsets of (X, τ ) is denoted by α(X, τ ) (or τ α ) [resp. PO (X, τ ); SO (X, τ ); SPO (X, τ )]. The family α(X, τ ) is a topology on X larger than τ , in general [12]. The complement of an α-open [resp. semiopen] subset of (X, τ ) is called α-closed [resp. semi-closed] in (X, τ ). The family of all closed [resp. semi-closed] subsets of (X, τ ) will be denoted by C (X, τ ) [resp. SC (X, τ )]. The intersection of all semi-closed subsets of (X, τ ) containing a certain set S ⊂ X is called semi-closure of S and is denoted by scl (S). Analogously, the semi-interior of S ⊂ X, denoted by sint (S), is the union of all semi-open subsets of (X, τ ) contained in S.The following equalities are also known [3, Theorem 1.6]:. The family of all regular open [resp. regular closed] subsets of (X, τ ) is denoted by RO (X, τ ) [resp. RC (X, τ )]. By definition we put R (X, τ ) = RO (X, τ ) ∩ RC (X, τ ). The class R (X, τ ) is the class of all clopen subsets of (X, τ ), that is R (X, τ ) = C (X, τ ) ∩ τ . The following known results will be useful in the sequel. By [1, Theorem 1.5(a), (b)] we have scl (S) = S ∪ int (cl (S)) and (resp.) sint (S) = S ∩ cl (int (S)). This theorem implies at once that scl (S) = int (cl (S)) iff S ∈ PO (X, τ ) [8, Proposition 2.7(a)]. A space (X, τ ) is extremally disconnected if cl (G) ∈ τ for each G ∈ τ . D. S. Janković proved, [7, Theorem 2.9], that (X, τ ) is e.d. iff SO (X, τ ) ⊂ τ α . By ∆ we designate the symmetric difference of two sets. Let us remark that by using of [17, Lemma 1], we can obtain another proof of this result. Before restating Theorem 1 in some ot...

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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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On some classes of nearly open sets

Cited by 718 publications

References 2 publications

α-ℽ-convergence, α-ℽ- accumulation and α-ℽ- compactness

α-ℽ-convergence, α-ℽ- accumulation and α-ℽ- compactness

On some classes of sets in extremally disconnected spaces

On almost precontinuous functions

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