Research article · 2016 · Full text
International Journal of Statistics and Applied MathematicsFuzzy pgprw-continuous maps and fuzzy pgprw-irresolute in fuzzy topological spaces
Volume 1(1) · Pages 01–07
Research areasFuzzy and Soft Set Theory
Abstract
Author-writtenIn this paper, we introduce the concept of fuzzy pgprw-continuous maps and fuzzy pgprw-irresolute maps in fts. We prove that the composition of two fuzzy pgprw-continuous maps need not be fuzzy pgprw-continuous and study some of their properties.
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
- Fuzzy subsets, introduced by L.A. Zadeh in 1965, serve as a generalization of ordinary subsets and have applications across various engineering and science fields. View evidence ↗
- Fuzzy topological spaces (fts) were introduced by C.L. Chang in 1968, extending the theory of topological spaces to fuzzy sets. View evidence ↗
The question
- The authors aim to introduce and investigate the properties of a new class of fuzzy sets known as fuzzy pgprw-closed sets within fuzzy topological spaces. View evidence ↗
- The study also seeks to define and analyze fuzzy pgprw-continuous and fuzzy pgprw-irresolute maps. View evidence ↗
What they did
- The authors define a fuzzy pgprw-continuous map as one where the inverse image of every fuzzy open set in the codomain is fuzzy pgprw-open in the domain. View evidence ↗
- A fuzzy pgprw-irresolute map is defined as a map where the inverse image of every fuzzy pgprw-open set in the codomain is fuzzy pgprw-open in the domain. View evidence ↗
What they found
- Every fuzzy continuous map is also a fuzzy pgprw-continuous map. View evidence ↗
- Any fuzzy almost continuous map is necessarily a fuzzy pgprw-continuous map. View evidence ↗
- A map can be fuzzy pgprw-continuous without being an almost continuous map. View evidence ↗
- A map that is both fuzzy pgprw-continuous and fuzzy completely semi-continuous is a fuzzy continuous map. View evidence ↗
- The composition of a fuzzy pgprw-continuous map and a fuzzy continuous map results in a fuzzy pgprw-continuous map. View evidence ↗
- Every fuzzy pgprw-irresolute map is also a fuzzy pgprw-continuous map. View evidence ↗
What it means
- The authors conclude that they have successfully introduced and investigated the properties of fuzzy pgprw-continuous and fuzzy pgprw-irresolute maps. View evidence ↗
- The authors state that future work will involve analyzing the same process for other fuzzy pgprw-properties. View evidence ↗
Visual summary
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Hierarchy of Fuzzy Continuity
AI-generated conceptual illustration (Abstract geometric shapes representing mathematical sets in a fuzzy topological space). It explains the idea and does not show study data.
View source evidence ↗Introduction
The concept of a fuzzy subset was introduced and studied by L.A. Zadeh in the year 1965. The subsequent research activities in this area and related areas have found applications in many branches of science and engineering. In the year 1965, L.A.Zadeh [1] . introduced the concept of fuzzy subset as a generalization of that of an ordinary subset. The introduction of fuzzy subsets paved the way for rapid research work in many areas of mathematical science. In the year 1968, C.L.Chang [2] introduced the concept of fuzzy topological spaces as an application of fuzzy sets to topological spaces . Subsequently several researchers contributed to the development of the theory and applications of fuzzy topology. The theory of fuzzy topological spaces can be regarded as a generalization theory of topological spaces. An ordinary subset A or a set X can be characterized by a function called characteristic function A : X [0,1] of A, defined by
Thus an element x in A if A (x) = 1 and is not in A if A (x) = 0. In general if X is a set and A is a subset of X then A has the following representation. A = { ( x , A (x)): x X}, here A (x) may be regarded as the degree of belongingness of x to A, which is either 0 or 1. Hence A is the class of objects with degree of belongingness either 0 or 1 only. Prof. L.A.Zadeh [1] introduced a class of objects with continuous grades of belongingness ranging between 0 and 1 ; he called such a class as fuzzy subset. A fuzzy subset A in X is characterized as a membership function A : X [0,1] , which associates with each point in x a real number A(x) between 0 and 1 which represents the degree or grade membership of belongingness of x to A. The purpose of this paper is to introduce a new class of fuzzy sets called fuzzy pgprw-closed sets in fuzzy topological spaces and investigate certain basic properties of these fuzzy sets. Among many other results it is observed that every fuzzy closed set is fuzzy pgprw-closed but not conversely. Also we introduce fuzzy pgprw-open sets in fuzzy topological spaces and study some of their properties.
1 on X and is denoted by x or Ix. In fact every subset of X is a fuzzy subset of X but not conversely. Hence the concept of a fuzzy subset is a generalization of the concept of a subset.
Defnition: [1]
If A and B are any two fuzzy subsets of a set X , then A is said to be included in B or A is contained in B iff A(x) ≤ B(x) for all x in X. Equivalently, A ≤ B iff A(x) ≤ B(x) for all x in X. [1] Two fuzzy subsets A and B are said to be equal if A(x) = B(x) for every x in X. Equivalently A = B if A(x) = B(x) for every x in X.
Defnition:
Defnition:
Defnition: [1]
The complement of a fuzzy subset A in a set X, denoted by A′ or 1 -A, is the fuzzy subset of X defined by A′(x) = 1 -A(x) for all x in X. Note that (A′)′ = A.
Defnition: [1]
The union of two fuzzy subsets A and B in X, denoted by A ∨ B, is a fuzzy subset in X defined by (A ∨ B)(x) = Max{ A(x), B(X)} for all x in X.
Defnition: [1]
The intersection of two fuzzy subsets A and B in X, denoted by A ∧ B, is a fuzzy subset in X defined by (A ∧ B)(x) = Min{A(x), B(x)} for all x in X.
Defnition: [1]
A fuzzy set on X is 'Crisp' if it takes only the values 0 and 1 on X.
Defnition: [2]
Let X be a set and be a family of fuzzy subsets of (X, is called a fuzzy topology on X iff satisfies the following conditions. (i) ϕ ; X ∈ : That is 0 and 1 ∈
The pair (X, ) is called a fuzzy topological space (abbreviated as fts). The members of are called fuzzy open sets and a fuzzy set A in X is said to be closed iff 1 -A is an fuzzy open set in X.
1.9 Remark: [2] Every topological space is a fuzzy topological space but not conversely.
Defnition: [2]
Let X be a fts and A be a fuzzy subset in X. Then ∧ {B : B is a closed fuzzy set in Xand B ≥ A} is called the closure of A and is denoted by A or cl(A).
Defnition: [2]
Let A and B be two fuzzy sets in a fuzzy topological space (X, )and let A ≥ B. Then B is called an interior fuzzy set of A if there exists G ∈ such that A ≥ G ≥ B, the least upper bound of all interior fuzzy sets of A is called the interior of A and is denoted by A 0 .
Definition: [3]
A fuzzy set A in a fts X is said to be fuzzy semiopen if and only if there exists a fuzzy open set V in X such that V A cl(V).
Definition: [3]
A fuzzy set A in a fts X is said to be fuzzy semi-closed if and only if there exists a fuzzy closed set V in X such that int(V) A V. It is seen that a fuzzy set A is fuzzy semiopen if and only if 1-A is a fuzzy semi-closed.
Theorem: [3]
The following are equivalent:
References
8 in the original paper- Zadeh LA. Fuzzy sets, Information and control, 1965; 8: 338-353.
- Chang CL. Fuzzy topological spaces, JI. Math. Anal. Appl., 1968; 24:182-190.
- Azad KK. On fuzzy semi continuity, fuzzy almost continuity and fuzzy weakly continuity. JI. Math. Anal. Appl. 1981; 82(1):14-32.
- Wali RS. On some topics in general and fuzzy topological spaces; ph. d thesis, Karnatak university Dharwad, 2007.
- Wali RS, Vivekananda Dembre. Fuzzy pgprw-closed sets and Fuzzy pgprw-open sets in Topological Spaces; journal of global research in Mathematics archives, 2016.
- Azad kk. On fuzzy semi-continuity, fuzzy almost continuity and fuzzy weak continuity, j. math. anal app 1981; 829:14-81.
- Mukerjii MN, Sinha SP. irresolute and almost open continuous between fts, fuzzy sets and systems, 1989; 29:381-388.
- Anjan Mukherjee. On fuzzy completely semi-continuous & weakly completely, semi-continuity functions, indian J pure Appl math, 1998; 29(2):191-197.
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Vivekananda Dembre · International Journal of Computer Applications Technology and Research · 2018
In this paper, we study some separation axioms namely, w-To-space, w-T1 -space and w-T2-space and their properties.We also obtain some of their characterizations.
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