On locally s-closed spaces
01 Jan 1996-International Journal of Mathematics and Mathematical Sciences (Hindawi)-Vol. 19, Iss: 1, pp 67-73
TL;DR: In this article, the concept of s-closed subspaces has been introduced and the following impo(t=nt ptopertles are establlshed Let X be an s-8-closed sur-3ectectlon wlth s-set (Malo and Nolrl 8i) polnt Inve(ses.
Abstract: in the prent papz, the concepts of s-closed sub-spaces, loCd(ly s- closed spaces have been lt(oduced and chardCt(Ized. We have seen that local s- c{osednss is seml-rgulat propetty; the concept ot s-8-closed mapping has been introduced here and the following impo(t=nt ptopertles are establlshed Let X --) be an s-8-closed sur3ectlon wlth s-set (Malo and Nolrl 8i) polnt Inve(ses. Then (d) it iS completely continuous (Ary and Gupta |i)) ad Y is locally compact q -space, then, X Is locally s-closed. g (b) It f is -conttnuous IGanguly and BdSU {5) and X s a locally compact T-
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TL;DR: In this paper, the authors studied the relation between semi-regular sets and semi-open sets and gave new characterizations of s-regular, s-normal, semi-Hausdorff, and s-normal topological spaces.
Abstract: A set A of a topological space (X,τ) is semi-θ-open if A is the union of semi-regular sets, i.e. sets which are both semi-open and semi-closed. Recently, several covering properties in terms of semi-θ-open sets have been introduced. The aim of this paper is to study the relativity of those properties with respect to arbitrary subsets. We give new characterizations of s-regular, s-normal, semi-Hausdorff and \(T_{\frac{3}{4}} \)-spaces.
2 citations
Posted Content•
TL;DR: In this paper, the authors investigated some common and controversial covering properties of topological spaces and their associated topology in both topological and bitopological ways, and proposed covering properties for the topology of a topological space and its associated α-topology.
Abstract: Recently, Mrsevic and Reilly discussed some covering properties of a topological space and its associated $\alpha$-topology in both topological and bitopological ways. The main aim of this paper is to investigate some common and controversial covering properties of $\cal T$ and ${\cal T}^{\alpha}$.
2 citations
Cites background from "On locally s-closed spaces"
...of A by semi-open sets in (X,T ), there exists a finite subfamily whose closures (resp. semi-closures) in (X,T ) from a cover of A. Furthermore, (X,T ) is locally S-closed [27] (resp. locally s-closed [2]) if each point of X has a neighbourhood which is S-closed relative to X (resp. s-closed relative to X). Corollary 2.6 Let (X,T ) be a topological space. Then: (a) (X,T α) is locally S-closed if and o...
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Journal Article•
TL;DR: In this paper, a characterisation of Hausdorff spaces in terms of s-closedness and compactness relative to a space, and relations between some of them are indicated.
Abstract: Some properties of sets s-closed or S-closed relative to a space, and s-closed or S-closed subspaces, are obtained. Relationships between some of them are indicated. New characterizations of Hausdorff spaces in terms of s-closedness and compactness relative to a space, are obtained.
1 citations
Cites background from "On locally s-closed spaces"
...In [4] the following two results have been stated....
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01 Jan 2003
TL;DR: In this article, a new class of functions called semi θ-perfect functions were defined and investigated, and the characterizations of locally s-closed spaces in weakly Hausdorff spaces were obtained.
Abstract: In this paper, we define and investigate a new class of functions called semi θ-perfect functions and also obtain the characterizations of locally s-closed spaces in weakly Hausdorff spaces.
01 Jan 2022
References
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TL;DR: In this article, a semi-Open Sets and Semi-Continuity in Topological Spaces (SOCS) model is proposed, which is based on the semi-continuity in topological spaces.
Abstract: (1963). Semi-Open Sets and Semi-Continuity in Topological Spaces. The American Mathematical Monthly: Vol. 70, No. 1, pp. 36-41.
1,630 citations
1,116 citations
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01 Feb 1980
TL;DR: A characterization of H-closed spaces in terms of projections is given in this article, which is an analogue to the following theorem for compact spaces: a space X is compact if and only if the projection from X x Y onto Y is a closed function for every space Y [9, p. 21].
Abstract: A characterization of H-closed spaces in terms of projections is given along with relating properties. Introduction. The primary purpose of this paper is to give a characterization of H-closed spaces which is an analogue to the following theorem for compact spaces: A space X is compact if and only if the projection from X x Y onto Y is a closed function for every space Y [9, p. 21]. Following the notation of [6], we utilize the notion of 9-closed subsets of a topological space from [ 11, p. 103] and our characterization is stated as follows: THEOREM. A Hausdorff space X is H-closed if and only if for every space Y, the projection from X x Y onto Y takes 9-closed subsets onto 9-closed subsets. Throughout, cl(K) will denote the closure of a set K. Preliminary definitions and theorems. DEFINITION 1. A net in a topological space is said to 9-converge (9accumulate) [6, Definition 3] to a point x in the space if the net is eventually (frequently) in cl( V) for each V open about x. DEFINITION 2. A point x in a topological space X is in the 9-closure [11, p. 103] of a set K c X (8-cl(K)) if cl(V) n K #0 for any V open about x. DEFINITION 3. A subset K of a topological space is 9-closed [11, p. 103] if it contains its 9-closure (i.e., 0-cl(K) c K). The following theorems give some parallels of properties of closure and closed sets in a topological space for 9-closure and 9-closed sets in the space and some relationships between these notions. The proofs of these theorems are straightforward and are omitted [11, Lemmas 1, 2, 3]. THEOREM 1. A point x in a topological space is in the 9-closure of a subset K if and only if there is a net xO in K which 9-converges to x (x,, 6 x). THEOREM 2. In any topological space (a) the empty set and the whole space are 9-closed, (b) arbitrary intersections and finite unions of 9-closed sets are 9-closed, (c) cl(K) c 0-cl(K) for each subset K, (d) a 9-closed subset is closed. Received by the editors June 30, 1975. AMS (MOS) subject classifications (1970). Primary 54D20.
191 citations
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01 Jan 1976
TL;DR: A topological space X is said to be S-closed if and only if for every semi-open cover of X there exists a finite subfamily such that the union of their closures cover X as discussed by the authors.
Abstract: A topological space X is said to be S-closed if and only if for every semiopen cover of X there exists a finite subfamily such that the union of their closures cover X. For a compact Hausdorff space, the concept of S-closed is shown to be equivalent to the concepts of extremally discon- nected and projectiveness.
122 citations
"On locally s-closed spaces" refers background in this paper
...set), then X is called s-closed [8] (resp. S-closed [ 14 ]). A subset A is called...
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19 citations
"On locally s-closed spaces" refers background in this paper
...[ 13 ] further generalzed S-closed spaces to locally S-closed spaces, in this paper...
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