Research article · 2018 · Full text
International Journal of Statistics and Applied MathematicsA study on soft GSR-closed sets in soft topological spaces
Volume 3(1) · Pages 259–262
Research areasFuzzy and Soft Set Theory
Abstract
Author-writtenIn this paper the concept of soft -gsr-closed and open sets in soft topological s paces are defined. The properties on soft closed set, soft regular open, soft-gsr-closed sets and soft-gsr open sets are discussed.
Summary
Source-checkedPlain-language reading of the full paper, prepared with AI and checked against the text. Each point links to the authors' own words.
Why it matters
- Soft set theory was introduced as a new mathematical tool by Molodtsov. View evidence ↗
- The foundations of soft topological spaces, including soft open and closed sets, were established by Shabir and Naz. View evidence ↗
- Generalized closed and open sets in topological spaces were originally introduced by Levine. View evidence ↗
The question
- The paper aims to define and explore the concepts of soft-gsr-closed and soft-gsr-open sets within soft topological spaces. View evidence ↗
What they did
- A soft-gsr-open set is defined as a soft set whose relative complement is a soft-gsr-closed set. View evidence ↗
What they found
- All soft closed sets are soft-gsr-closed, but not all soft-gsr-closed sets are soft closed. View evidence ↗
- Every soft g-closed set is also soft-gsr-closed, though the reverse is not necessarily true. View evidence ↗
- Soft α-closed sets are always soft-gsr-closed, but soft-gsr-closed sets are not always soft α-closed. View evidence ↗
- Soft semi-closed sets are a subset of soft-gsr-closed sets, but the converse does not hold. View evidence ↗
- Every soft rg-closed set is soft-gsr-closed, but not every soft-gsr-closed set is soft rg-closed. View evidence ↗
- Soft πgr-closed sets are always soft-gsr-closed, but the converse is not always true. View evidence ↗
- A soft set that is both soft regular open and soft-gsr-closed is necessarily soft semi-closed. View evidence ↗
- For a soft-gsr-closed set, the difference between its soft semi-closure and the set itself contains only the null soft regular closed set. View evidence ↗
- A soft-gsr-closed set is soft semi-closed if and only if the difference between its soft semi-closure and itself is soft regular closed. View evidence ↗
- The property of being soft-gsr-open is preserved under the intersection of two such sets. View evidence ↗
What it means
- The authors conclude that soft set theory is a rich field for both theoretical and experimental investigation. View evidence ↗
- The authors suggest that future work should focus on introducing new parameters into soft closed and soft open sets. View evidence ↗
Visual summary
1 visualsIllustrations explain the paper; they do not show study data.
Hierarchy of Soft Closed Sets
AI-generated conceptual illustration (Abstract geometric nested spheres representing mathematical set hierarchies). It explains the idea and does not show study data.
View source evidence ↗Preliminaries Definition: 2.1
Let U be the initial universe, P(U) be the power set of U and E be the set of all parameters Let A be a non -empty subset of E. A pair (F, A) is called a soft set over U, where F is a mapping given by F: A→ P(U). In other words, a soft set over U is a parameterized family of subsets of the universe U. F (ℇ) may be considered as the set ℇapproximate elements Of the soft set (F, A) for ℇ 𝝐 A. For two soft sets (F,A) and (G,B) over the common universe U, (F,A) is a soft subset of (G,B) if (i) A B and (ii) for all e 𝝐 A, F (e) and G(e) are identical approximations. We write (F, A)
Two soft sets (F, A) and (G, B) over a common universe U are said to be soft equal if (F, A) is a soft subset of (G, B) and (G, B) is a soft subset of (F, A).
Definition: 2.2
The union of two soft sets of (F, A) and (G, B) over the common universe U is soft set (H, C), where C = A U B and for all e 𝝐 C (i) H(e) = F(e) if e 𝝐 A-B, (ii) H(e) = G(e) if e 𝝐 B-A and (iii) H(e) = F(e) U G(e) if e 𝝐 A ∩ B, (F, A) U (G, B) = (H,C).
Definition: 2.3
The Intersection (H, C) of two soft sets (F, A) and (G,B) over a common universe U denoted (F,A) ∩ (G,B) is defined as C = A∩B and H (e) = F (e) ∩ G (e) for all e 𝝐 C.
Definition: 2.4
For a soft set (F, A) over the universe U, the relative complement of (F, A) is denoted by (F,A) c and is defined by (F, A) c = (F c , A), where F c : A → P(U) is a mapping defined by F c (e) = U -F(e) for all e 𝝐 A.
Definition: 2.5
Let τ be the collection of soft sets over X, then τ is called a soft topology on X if τ satisfies (i) ф, X belong to τ.
(ii) The union of any number of soft sets in τ belongs to τ.
(iii)The intersection of any two soft sets in τ belongs to τ. The triplet (X, τ, E) is called a soft topological s pace over X and it is denoted by X (For simplicity).
Definition: 2.6
Let (X, τ, E) be soft s pace over X. A soft set (F, E) over X is said to be soft closed in X, if its relative complement (F, E) c belongs to τ. The relative complement is a mapping F c : E→ P(X) defined by F c (e) = X -F(e) for a ll e 𝝐 A.
Definition: 2.7
Let X be an initial universe set, E be the set of parameters and τ = {ф, X}. Then τ is called the soft indiscrete topology on X and (X, τ, E) is said to be a soft indiscrete s pace over X. If τ is the collection of all soft sets defined over X, then τ is called the soft discrete topology on X and (X, τ, E) is said to be a soft discrete s pace over X.
Definition
Definition
Definition: 2.10
Definition: 2.10
Definition: 2.11
The soft semi closure of (A, E) is the intersection of all soft semi closed sets containing (A, E). (i.e) The smalles t soft semi closed set containing (A,) and is denoted by sscl (A, E). The soft semi interior of (A,E) is the union of all soft semi open set contained in (A, E) and is denoted by ss int(A, E). Similarly, we define soft regular-closure, soft α-closure, soft pre-closure, soft semi closure and soft b -closure of the soft set (A, E) of a topological s pace X and are denoted by srcl (A, E), sαcl (A, E), spcl(A, E), sscl (A, E) and Sbcl (A, E) respectively. The ore m: 3.1 Every soft closed set is soft-gsr-closed, converse is not true.
Definition
Definition
References
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