Fuzzy Sets, Fuzzy S-Open and S-Closed Mappings

Several properties of fuzzy semiclosure and fuzzy semi-interior of fuzzy sets deﬁned by Yalvac (1988), have been established and supported by counterexamples. We also study the characterizations and properties of fuzzy semi-open and fuzzy semi-closed sets. Moreover, we deﬁne fuzzy s-open and fuzzy s-closed mappings and give some interesting characterizations.


Introduction
The concept of fuzzy set was introduced by Zadeh in his classical paper [1].This concept provides a natural foundation for treating mathematically the fuzzy phenomena, which exist pervasively in our real world, and for building new branches of fuzzy mathematics.In the area of Fuzzy Topology, introduced by Chang [2], much attention has been paid to generalize the basic concepts of General Topology in fuzzy setting and thus a modern theory of Fuzzy Topology has been developed.
In recent years, Fuzzy Topology has been found to be very useful in solving many practical problems.Du et al. [3] fuzzified the very successful 9-intersection Egenhofer model [4,5] for depicting topological relations in Geographic Information Systems (GIS) query.In [6,7], El Naschie showed that the notion of Fuzzy Topology might be relevant to quantum particle physics and quantum gravity in connection with string theory and e ∞ theory.Tang [8] used a slightly changed version of Chang's fuzzy topological space to model spatial objects for GIS databases and Structured Query Language (SQL) for GIS.
Levine [9] introduced the concepts of semi-open sets and semicontinuous mappings in topological spaces.Interestingly, his work found applications in the field of Digital Topology [10].For example, it was found that the digital line is a T 1/2 -space [11], which is a weaker separation axiom based upon semi-open sets.Fuzzy Digital Topology [12] was introduced by A. Rosenfeld, which demonstrated the need for the fuzzification of weaker forms of notions of Classical Topology.Azad [13] carried out this fuzzification in 1981, and presented some general properties of fuzzy spaces.Several properties of fuzzy semi-open (resp., fuzzy semi-closed), fuzzy regular open (resp., closed) sets have been discussed.Moreover he defined fuzzy semicontinuous (resp., semi-open and semi-closed) functions and studied the properties of fuzzy semicontinuous function in product related spaces.Finally, he defined and characterized fuzzy almost continuous mappings.For related subsequent work in this direction, we refer to [14][15][16][17][18][19][20][21][22][23][24][25][26][27].
In this paper, our aim is to further contribute to the study of fuzzy semi-open and fuzzy semi-closed sets defined by Yalvac [26] by establishing several important fundamental identities and inequalities supported by counterexamples.Cameron and Woods [28] introduced the concepts of s-continuous mappings and s-open mappings.They investigated the properties of these mappings and their relationships to properties of semi-open sets.Khan and Ahmad [29] further worked on the characterizations and properties of s-continuous, s-open and s-closed mappings.We fuzzify the findings of [28,29].We define fuzzy s-open and fuzzy s-closed mappings and establish some interesting characterizations of these mappings.

Preliminaries
In order to make this paper self-contained, we briefly recall certain definitions and results; for those not described, we refer to [1,2,13,26].

Advances in Fuzzy Systems
Let X = {x} be a space of points (objects), with a generic element x.A fuzzy set λ in X is characterized by membership function λ(x) from X to the unit interval [0, 1].
The symbol Φ denotes the empty fuzzy set defined as μ Φ (x) = 0, for all x ∈ X.For X, the membership function is defined as μ X (x) = 1, for all x ∈ X. Definition 1 (see [2]).Let f : X → Y be a mapping.Let β be a fuzzy set in Y with membership function β(y).Then the inverse of β, written as f −1 (β), is a fuzzy set in X whose membership function is defined by ( Conversely, let λ be a fuzzy set in X with membership function λ(x).The image of λ, written as f (λ), is a fuzzy set in Y whose membership function is given by for all y ∈ Y .

Every member of τ is called τ-open fuzzy set (or simply an open fuzzy set). A fuzzy set is τ-closed if and only if its complement is τ-open.
As in general topology, the indiscrete fuzzy topology contains only Φ and X, while the discrete fuzzy topology contains all fuzzy sets.
The class of all fuzzy semi-open (resp., fuzzy semi-closed) sets in X is denoted by FSO(X) (resp., FSC(X)).
Definition 4 (see [26]).Let λ be a fuzzy set in an fts X.Then semi-closure (briefly sCl) and semi-interior (briefly sInt) of λ are defined as and are called the fuzzy semi-closure of λ and fuzzy semiinterior of λ, respectively.
The following are characterizations of fuzzy semi-closed sets, the proof of Theorem 1 is straightforward.

Theorem 2. For a fuzzy set λ in an fts X, λ is fuzzy semi-closed if and only if sIntsCl
(⇐) Suppose sInt(sCl λ) ≤ λ.Since sCl λ is fuzzy semiclosed, so there exists a closed set ψ such that Int Hence λ is fuzzy semi-closed and λ = sCl λ.
The inequalities (1) and (4) of Theorem 3, are in general irreversible, as is shown by following.
We use this and Theorem 1, and prove the following.
Proof.(1) By the fact that sCl λ is fuzzy semi-closed and that λ is fuzzy semi-closed if and only if sCl λ = λ, it follows immediately.
Theorem 5.For any fuzzy set λ in an fts X, we have (sCl λ) c = sInt(λ c ). Proof. (1) (2) Similar to (1).Definition 6 (see [31]).A fuzzy point e is called a boundary point of a fuzzy set λ if and only if e ∈ Cl λ ∧ Cl λ c .The union of all the boundary points of λ is called a boundary of λ, denoted by Bd λ.It is clear that

Fuzzy S-Open and Fuzzy S-Closed Mappings
Next, we define Definition 7. Semiboundary (briefly sBd) of a fuzzy set λ in an fts X is defined as In the following, we characterize fuzzy s-open mappings in terms of sInt, sCl, and sBd.Theorem 6.For a function f : X → Y , a fuzzy set α in an fts X and a fuzzy set β in an fts Y , then the following are equivalent: is the largest fuzzy open set such that Int f (α) ≤ f (α).Therefore f (sInt α) ≤ Int f (α), for any fuzzy set α in X.This gives (2).
In the following, we give characterizations of fuzzy sclosed mappings as follows.
This proves that f (λ) is fuzzy closed.Proof.Let ψ be an arbitrary fuzzy semi-closed set in X and y ∈ ( f (ψ)) c .Then

First, we define Definition 5 .
A function f : X → Y is said to be fuzzy sopen (resp., fuzzy s-closed) if the image of every fuzzy semiopen (resp., fuzzy semi-closed) set is fuzzy open (resp., fuzzy closed).Obviously a fuzzy s-open function is fuzzy open.

Theorem 8 . 15 ) 9 .
If a function f : X → Y is fuzzy s-closed then for each fuzzy set β in an fts Y and each fuzzy semi-open set μ in an fts X with μ ≥ f −1 (β), there exists a fuzzy open set ν in Y with ν≥ β such that f −1 (ν) ≤ μ.Proof.Let μ be an arbitrary fuzzy semi-open set in X with μ ≥ f −1 (β), where β is a fuzzy set in Y .Clearly ( f (μ c )) c = ν (say) is fuzzy open in Y .Since f −1 (β) ≤ μ, then straightforward calculations give that β ≤ ν.Moreover, we have f −1 (ν) = f −1 ( f (μ c )) c = f −1 f (μ c ) c ≤ μ or f −1 (ν) ≤ μ. (Theorem Let f : X → Ybe a surjective function from an fts X to an fts Y.If for each fuzzy set β in Y and each fuzzy semi-open set μ in X with μ ≥ f −1 (β), there exists a fuzzy open set ν in Y with ν ≥ β such that f −1 (ν) ≤ μ, then f is s-closed.
) or f −1 (y) ≤ ψ c .Since ψ c is fuzzy semi-open, therefore there exists a fuzzy open set ν y with y∈ ν y such that f −1 (ν y ) ≤ ψ c .Since f is surjective, we have y ∈ ν y ≤ ( f (ψ)) c .Thus ( f (ψ)) c = ∨{ν y | y ∈ ( f (ψ)) c } is fuzzy open in Y or f (ψ) is fuzzy closed in Y .This proves that f is s-closed.Combining Theorems 8 and 9, we have the following.Theorem 10.A surjective function f : X → Y is fuzzy sclosed if and only if for each fuzzy set β in Y and each fuzzy semi-open set μ in X with μ ≥ f −1 (β), there exists a fuzzy open set ν in Y with ν ≥ β such that f −1 (ν) ≤ μ.