On semi-regularization topologies

Mršević, Mila; Reilly, Ivan L.; Vamanamurthy, M. K.

doi:10.1017/s1446788700022588

Cited by 51 publications

(34 citation statements)

References 12 publications

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“…It turns out that using δ-open sets is another way to describe the semi-regularization topology, that is, τ δ = τ s for any space (X, τ ). Semiregularization topologies are considered in some detail by Mrsevic, Reilly and Vamanamurthy [8], especially from the change of topology perspective.…”

Section: Introductionmentioning

confidence: 99%

Generalization of perfectly continuous, regular set-connected and clopen functions

Ekici

2005

Acta Math Hung

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Abstract. Recently the class of almost cl-supercontinuous functions between topological spaces has been studied in some detail. We consider this class of functions from the point of view of change(s) of topology. In particular, we conclude that this class of functions coincides with the usual class of continuous functions when the domain and codomain have been retopologized appropriately. Some of the consequences of this fact are considered in this paper. AMS Classification:Primary 54C60, 54C05, 54C10; Secondary 54D10, 54D20.

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Section: Introductionmentioning

confidence: 99%

Generalization of perfectly continuous, regular set-connected and clopen functions

Ekici

2005

Acta Math Hung

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“…Note that the topological analogue of the result above is Proposition 12 (1) of Mršević, Reilly and Vamanamurthy [15].…”

Section: Change Of Generalized Topologymentioning

confidence: 70%

“…Recall that a subset B of (X, τ ) is defined to be τ -regular open if B = i(c(B)). Mršević, Reilly and Vamanamurthy [15] have given a systematic discussion of this relationship, especially from the point of view of change of topology. On the other hand, Veličko [18] provided the basic discussion of the properties of θ-topologies.…”

Section: Six Classes Of Generalized Topologiesmentioning

confidence: 99%

On g-α-irresolute functions

Bai

Zuo

2010

Acta Math Hung

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“…Mrsevic, Reilly and Vamanamurthy [6] and Saleemi, Shahzad and Alghamdi [9] studied this notion and made their contribution to the subject. The aim of this paper is to prove a general preservation theorem for almost locall connectedness and to generalize the above result of Jankovic [3].…”

Section: Let F : X -• Y Be a Weakly Continuous Almost Open Surjectiomentioning

confidence: 99%

Remarks on Almost Locally Connected Spaces

Shahzad¹,

Al-Malki²

2006

Demonstratio Mathematica

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Abstract. Some preservation theorems for almost local connectedness are proved. IntroductionIn 1981, as a generalization of locally connected topological spaces, the concept of almost locally connected spaces was introduced and studied by Mancuso [5]. Subsequently, Noiri [8] extended the work of Mancuso. In 1985, Jankovic [3] generalized and improved Noiri's results and among others, obtained the following result: Let f : X -• Y be a weakly continuous, almost open surjection. If X is almost locally connected, then so is Y.Mrsevic, Reilly and Vamanamurthy [6] and Saleemi, Shahzad and Alghamdi [9] studied this notion and made their contribution to the subject. The aim of this paper is to prove a general preservation theorem for almost locall connectedness and to generalize the above result of Jankovic [3]. PreliminariesThroughout this paper spaces mean topological spaces on which no separation axioms are assumed unless explicitly stated. Let A be a subset of a space X. The closure of A and the interior of A are denoted by Cl(A) and Int(A), respectively. The set A is called regular open (resp. regular closed) if Int(Cl(A)) = A (resp. Cl(Int{A)) = A). The family of all regular open subsets of a space X is represented by RO(X). The family RO(X) is a basis for a topology on X and the set X with this topology is called the semi-regularization of X and is denoted by X s . A space X is called almost locally connected at a; € A" if for each G G RO(X) containing x there exists an open connected set V such that x G V C G. X is almost locally connected if it is almost locally connected at each of its points. Every locally connected space is almost locally connected; however the converse is not true Note that continuity => almost continuity =>• closure continuity => weak continuity and none of these implications is reversible. For any AC X, we define : for each open set U in X. Clearly every almost open surjection is s-almost open. But the converse is false [9]. Main resultsWe shall need the following results in the sequel. LEMMA 3.1 ([11]). Let f : X -> Y be a connected function and C be a component ofY. Then / -1 (C) is a union of some connected components of X. LEMMA 3.2 ([8]). A space X is almost locally connected if and only if the components of regular open sets in X are regular open.Now we prove our main results. If X is almost locally connected, then so isY. Proof. Let V € RO(Y) and C a component of V. Then by Lemma 3.1, f~l(C) is a union of some components of / -1 (F). As in [11], / -1 (C) fl Int{Cl(f~l(V)))is a union of some components of Int{Cl{f~l{V))). Since X is almost locally connected, it follows that f~l(C) fl Int{Cl(f~l{V))) is UyeC, then / _1 (y) C / _1 (C). Since C C V, by condition (S) we have that f-\y) Q Int{f~l{V)).Thus /-%) C f-\C)f\Int{Cl{r\V))) and so C C f*(f-\C)nlnt(Cl(f-\V)))).On the other hand,

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On semi-regularization topologies

Cited by 51 publications

References 12 publications

Generalization of perfectly continuous, regular set-connected and clopen functions

Generalization of perfectly continuous, regular set-connected and clopen functions

On g-α-irresolute functions

Remarks on Almost Locally Connected Spaces

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