On induced map f# in fuzzy set theory and its applications to fuzzy continuity


 In this paper, we introduce # image of a fuzzy set which gives a induced map f # corresponding to any function f : X → Y , where X and Y are crisp sets. With this, we present a new vision of studying fuzzy continuous mappings in fuzzy topological spaces where fuzzy continuity explains the term of closeness in the mathematical models. We also define the concept of fuzzy saturated sets which helps us to prove some new characterizations of fuzzy continuous mappings in terms of interior operator rather than closure operator.

and generalized the notion of continuity to what he named fuzzy continuous (F-continuous) functions and studied some of its characterizations. To introduce fuzzy continuity, he introduced image and inverse image of a fuzzy set corresponding to any function f : X → Y and established some properties of fuzzy sets induced by mappings.
In [4], Ming et. al. introduced the concept of closure and interior of a fuzzy set in a fuzzy topological space and studied some of its properties which are similar to that in general topological space. They also studied some more characterizations of fuzzy continuous functions in their other paper [5]. In [2], Arkhangel'skii introduced a new map f # : ℘(X) → ℘(Y ) given by f # (E) = {y ∈ Y |f −1 (y) ⊂ E} for any subset E of X, where ℘(X) is a collection of all subsets of X. This gives us the idea about the possibility of introducing such type of induced map using fuzzy subsets and use it to find possible characterizations of fuzzy continuity.
In this paper, apart from image and inverse image of a fuzzy set we introduce # image of a fuzzy set. Equivalently, given any function f : X → Y , where X and Y are crisp sets, it gives rise to a induced map f # : F Z(X) → F Z(Y ), where by F Z(X), we mean collection of all fuzzy subsets of X (Definition 7). We prove the monotonicity of map f # along with its various properties (Lemma 1). Relationship between # image of a fuzzy set and image of a fuzzy set is also given (Lemma 2). Further, we define fuzzy saturated sets and obtain its characterizations. It is also shown that surjective maps can be characterized in terms of # image of a fuzzy set (Theorem 4). Finally, using f # map, we give characterizations of fuzzy continuous maps in terms of interior operator rather than closure operator (Theorem 5) and using saturated sets (Theorem 6).

Preliminaries
In this section we will give some basic definitions and results that we need in our further sections. Let X = {x} be a space of points then a fuzzy set E in X is a function µ E from X to closed interval [0, 1] i.e. µ E : X → [0, 1]. In other words, a fuzzy set is characterized by a membership function which associates with each x in X its "grade of membership" µ E (x), in E. An empty fuzzy set, denoted by φ is defined as µ φ (x) = 0 for all x in X and fuzzy set X is defined as µ X (x) = 1 for all x in X. For a fuzzy set E, is known as height of E and a fuzzy set E is said to be normalized if and only if Hgt(E) = 1. Also, we have Definition 1 [3] "Let A and B be fuzzy sets in a space X = {x} with membership functions µ A (x) and µ B (x) respectively. Then Definition 2 [3] "A family T of fuzzy sets in X is said to be a fuzzy topology if it satisfies following conditions: Every member of T is called a T -open fuzzy set and the pair (X, T ) is known as fuzzy topological space or fts for short. Also a fuzzy set is T -closed if and only if its complement is T -open." Definition 3 [4] "Let (X, T ) be fuzzy topological space and E be any fuzzy set in X. Then the union of all T -open fuzzy sets contained in E is called Definition 4 [4] "Let (X, T ) be fuzzy topological space and E be any fuzzy set in X. Then the intersection of all T -closed fuzzy sets containing E is called the closure of E, denoted by E. Clearly, E is the smallest T -closed fuzzy set containing E and E = E." Theorem 1 [4] "In a fuzzy topological space (X, T ), for any fuzzy subset E of X, Definition 5 [3] "Let f : X → Y be a function and E be a fuzzy set in X with membership function µ E (x) then the image of E, denoted by f (E) is a fuzzy set in Y whose membership function is defined by, Also for a fuzzy set B in Y with membership function µ B (y), inverse image of B, denoted by f −1 (B) is a fuzzy set in X whose membership function is defined by for all x in X." Theorem 2 [3] "Let f : X → Y be any function. Then for any fuzzy subsets E, F in X and B in Y , the following holds : where g • f is the composition of g and f ." 3 Induced map f # We begin by introducing # image of a fuzzy set which gives rise to the induced Definition 7 Let X = {x} and Y = {y} be spaces of points. Let f be a function from X to Y and E be a fuzzy set in X with membership function µ E (x). Then # image of a fuzzy set E, written as f # (E), is a fuzzy set in Y with membership function defined by Note : From the definition of f # map, it can be seen easily that # image of any fuzzy set is normalized fuzzy set for any map f : X → Y , which is not surjective. But if f is surjective then # image of a fuzzy set need not be normalized fuzzy set.
Definition 8 Let f : X → Y be a function. Define a fuzzy set The following Lemma gives some properties of f # map.
Lemma 1 Let f : X → Y be any function and E, F be fuzzy subsets of X and B be fuzzy subset of Y . Then 1 (B)). In particular, it can be seen easily that if f is surjective map then As we have seen in Lemma 1(e) that intersection is preserved by f # map. But the following Example shows that union need not be preserved by f # map.
The following Lemma gives some more properties of f # map which we shall need to characterize fuzzy continuity.
Lemma 2 Let f : X → Y be any function and E be any fuzzy subset of X and B be any fuzzy subset of Y . Then Proof (a) Firstly, let f −1 (B) ⊂ E then using Lemma 1(a), it follows that (b) Let f : X → Y and g : Y → Z be any two maps then g • f : X → Z. Therefore, for each z ∈ Z, we have Now, three cases arise : It follows that The following two Lemmas characterize fuzzy saturated sets in terms of image and # image of a fuzzy set.

Lemma 4
For any map f : X → Y and any fuzzy subset E in X, E is fuzzy saturated if and only if E = E # .
Proof Let E be any fuzzy saturated subset in X then Conversely, let E = E # then by Remark 2, it follows that E is fuzzy saturated subset of X.
The following result gives a new characterization of surjective maps using # image of a fuzzy set. Proof For each y ∈ Y , f −1 (y) = φ, since f is surjective. Therefore, which is possible if and only if f −1 (y) = φ for each y ∈ Y i.e. if and only if f is surjective, since by definition Equivalently, we can say that f is surjective if and only if f # (φ) = φ.
4 Applications of f # to fuzzy contnuity The following theorem gives the characterization of fuzzy continuous maps in terms of interior in fuzzy topology.
Theorem 5 Let f be a function from X to Y where X = {x} and Y = {y} be spaces of points then f is fuzzy continuous if and only if for each fuzzy subset The following corollary characterizes fuzzy continuity for surjective maps.
Corollary 1 Let f : X → Y be any surjective map. Then f is fuzzy continuous if and only if for each fuzzy subset E of X, (f (E # )) o ⊂ f ((E o ) # ).
Proof Proof follows from Theorems 5 and 4.
The following theorem gives equivalent conditions for a surjective fuzzy continuous map.
Theorem 6 Let f : X → Y be any surjective map. Then the following are equivalent.

Conclusion
Topological spaces not only have applications in mathematics but in various other fields like physics, geographical systems, computer, chemistry etc. But because of uncertainties and incomplete information of an element, it is difficult to apply the concept of topology in all real life problems. To overcome this difficulty, fuzzy sets and hence fuzzy topological spaces were introduced which deals with uncertainty of an object. In various real life applications of fuzzy set theory, fuzzy functions play an important role especially in fuzzy control and approximate reasoning. Motivated by the applications of fuzzy functions, we have introduced a new type of fuzzy function denoted by f # and have discussed their properties. Further, it motivates to contribute in the theoretical study on fuzzy continuous functions. We can also study fuzzy open and fuzzy closed functions using f # which will be our next target and to study fuzzy topological properties preserved by these maps.

Funding
No funding was received for conducting this study.