On Fuzzy Sp-Open Sets

A new class of generalized fuzzy open sets in fuzzy topological space, called fuzzy sp-open sets, are introduced, and their properties are studied and the relationship between this new concept and other weaker forms of fuzzy open sets we discussed. Moreover, we introduce the fuzzy sp-continuous (resp., fuzzy sp-open) mapping and other stronger forms of sp-continuous (resp., fuzzy spopen) mapping and establish their various characteristic properties. Finally, we study the relationships between all these mappings and other weaker forms of fuzzy continuous mapping and introduce fuzzy sp-connected. Counter examples are given to show the noncoincidence of these sets and mappings.


Introduction
In 1996, Dontchev and Przemski, [1] have introduced the concept of sp-open sets in general topology.In this paper, we extend the notion of sp-open sets to fuzzy topology space and study some notions based on this new concept.We further study the relation between fuzzy sp-open sets and other types of fuzzy open sets.We also introduce the concepts of fuzzy sp-continuous (resp., fuzzy sp-open) mapping, other stronger forms of fuzzy sp-continuous (resp., fuzzy sp-open) mapping, and discuss their relation with other weaker forms of fuzzy continuous mapping.

Preliminaries
Throughout this paper, by (X, τ) or simply by X we mean a fuzzy topological space (fts, shorty) and f : X → Y means a mapping f from a fuzzy topological space X to a fuzzy topological space Y .If u is a fuzzy set and p is a fuzzy singleton in X, then N(p), Int λ, cl u, u c denote, respectively, the neighborhood system of p, the interior of u, the closure of u, and complement of u.Now, we recall some of the basic definitions and results in fuzzy topology.Definition 2.1 (see [2]).A fuzzy singleton p in X is a fuzzy set defined by: p(x) = t, for x = x 0 and p(x) = 0 otherwise, where 0 < t ≤ 1.The point p is said to have support x 0 and value t.Definition 2.2.A fuzzy set u in a fts X is called fuzzy α-open [3] (resp., Fuzzy preopen [4], Fuzzy β-open [5] The family of all fuzzy α-open (resp., fuzzy preopen, fuzzy βopen, fuzzy semiopen) sets of X is denoted by FαO(X) (resp., FPO(X), FβO(X), FSO(X)).[4]).Let u be any fuzzy set.Then,

Fuzzy SP -Open Set
Clearly, τ is a fuzzy topology on X, and by easy computation, it follows that v 3 and v 4 are fuzzy sp-open sets.But (v 3 ∧ v 4 ) is not a fuzzy fuzzy sp-open set.
Theorem 3.6.For any fuzzy subset u of a fuzzy space X, the following properties are equivalent: (2) The class of all fuzzy sp-closed sets in X will be denoted be FSP-C(X).
Definition 3.8.Let u any fuzzy set.Then, By using Definitions 3.1, 3.7, and 3.8, we can prove the following theorems.
Theorem 3.9.Let u and v be the fuzzy sets in fts X.Then, the following statements hold Theorem 3.10.For a fuzzy subset λ of a fuzzy space X, the following statements are holding: Theorem 3.11.For any fuzzy subset u of a fuzzy space X, the following statements are equivalent:  Example 3.14.Let X = {a, b, c} and v 1 , v 2 , v 3 , and v 4 be fuzzy sets of X defined as Let Clearly, τ is a fuzzy topological space on X, and by easy computation, we can see: Clearly, τ is a fuzzy topological space on X, and by easy computation, it follows that v 3 is fuzzy β-open set which is not fuzzy sp-open.

Fuzzy SP -Continuous Mapping
Definition 4.1.A mapping f : (ii)⇒(iii) Let fuzzy singleton p be in X and every fuzzy open set v be in Y such that f (p) ⊆ v, there exists a fuzzy spopen u such that p ≤ u and u ≤ f −1 (v).So, we have p ≤ u and f (u (iii)⇒(i) Let v be a fuzzy open set in Y and let us take Remark 4.4.If f : X → Y is fuzzy sp-continuous mapping and g : Y → Z is fuzzy sp-continuous mapping, then go f : X → Z may not be a fuzzy sp-continuous mapping; this can be show by the following example.
Example 4.5.Let X = {a, b, c} and v 1 , v 2 , v 3 , v 4 and v 5 be fuzzy sets of X defined as, Consider, ftsτ 1 , τ 2 , and τ 3 where Proof.Let v be a fuzzy set of Z.Then, (go f . And because g is fuzzy continuous this implies that g −1 (v) is a fuzzy open set of Y and hence f −1 (g −1 (v)) is a fuzzy sp-open set in X.Therefore, go f is a fuzzy spcontinuous mapping.
From Definitions 4.1 and 4.3, we can have the above "Implication Figure 2" illustrates the relation between different classes of fuzzy sp-continuous (fuzzy semi sp-open) mappings.
The above "Implication Figure 3" illustrates the relation between fuzzy sp-continuous and different classes of fuzzy continuous mapping.Remark 4.7.We can see the converse of these relations need not be true, in general as shown by the following examples.
Example 4.8.Let X = {a, b, c} and v 1 , v 2 , v 3 , and v 4 fuzzy sets of X defined as Consider fts τ 1 , τ 2 , and τ 3 where Definition 4.9.A fuzzy set u in an fts X is said to be fuzzy connected if u cannot be expressed as the union of two fuzzy separated sets.Now, we can generalize the definition of fuzzy connected to define fuzzy sp-connected as follows.
Definition 4.10.A fuzzy set v in a fts (X, τ) is said to be fuzzy sp-connected if and only if v cannot be expressed as the union of two fuzzy sp-separated sets.Theorem 4.11.Let f : X → Y be a fuzzy sp -continuous surjective mapping.If v is a fuzzy sp -connected subset in X then, f (v) is fuzzy connected in Y .
Proof.Suppose that f (m) is not connected in Y .Then, there exist fuzzy separated subsets u and v in Y such that f (m) = u ∪ v.
Since f is fuzzy sp-continuous surjective mapping, f −1 (u) and f −1 (v) are fuzzy sp-open set in X and It is clear that f −1 (u) and f −1 (v) are fuzzy sp-separated in X.Therefore, m is not fuzzy sp-connected in X, which is a contradiction!! Hence, Y is fuzzy connected.

Figure 1 Proof.
Figure 1 fuzzy sp-open set which is neither fuzzy α-open set nor fuzzy preopen, (ii) v 3 is fuzzy sp-open which is not semiopen, (iii) v 4 is fuzzy sp-open set which is not fuzzy open.Example 3.15.Let X = {a, b, c} and v 1 , v 2 , and v 3 fuzzy sets of X defined as Proof.(i)⇒(ii) Let fuzzy singleton p be in X and every open set v in Y such that f (p) ⊆ v, there exists a fuzzy open set m be in Y such that f

Proposition 3.2. Let
Definition 3.1.A fuzzy subset u of fuzzy space X is called fuzzy sp-open set if u ≤ Int cl u ∨ cl Int u.The class of all fuzzy sp-open sets in X will be denoted be FSP − O(X).u be fuzzy sp -open set such that Int u = 0.Then, u is fuzzy preopen.
is fuzzy open set in X for each fuzzy sp-open set v in Y .
(ii) for every fuzzy singleton p in X and every open set v in Y such that f (p) ⊆ v, there exists a fuzzy sp -open set u ⊆ X such that p ⊆ u and u ≤ f −1 (v); (iii) for every fuzzy singleton p in X and every open set v in Y such that f (p) ⊆ v, there exists a fuzzy sp -open set u ⊆ X such that p ⊆ u and f (u) ≤ v; (iv) the inverse image of each fuzzy closed set in Y is fuzzy sp-closed;