Research Article Fuzzy Boundary and Fuzzy Semiboundary

We present several properties of fuzzy boundary and fuzzy semiboundary which have been supported by examples. Properties of fuzzy semi-interior, fuzzy semiclosure, fuzzy boundary, and fuzzy semiboundary have been obtained in product-related spaces. We give necessary conditions for fuzzy continuous (resp., fuzzy semicontinuous, fuzzy irresolute) functions. Moreover, fuzzy continuous (resp., fuzzy semicontinuous, fuzzy irresolute) functions have been characterized via fuzzy-derived (resp., fuzzy-semiderived) sets.


INTRODUCTION
After Zadeh's [1] introduction of fuzzy sets, Chang [2] defined and studied the notion of a fuzzy topological space in 1968.Since then, much attention has been paid to generalize the basic concepts of classical 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 semiopen 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 digital line is a T 1/2space [11], which is a weaker separation axiom based upon semiopen sets.Fuzzy digital topology [12] was introduced by 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 semiopen (resp., fuzzy semiclosed), fuzzy regular open (resp., closed) sets have been discussed.Moreover, he defined fuzzy semicontinuous (resp., semiopen, semiclosed) functions and studied the properties of fuzzy semicontinuous function in product-related spaces.Finally, he defined and characterized fuzzy almost continuous mappings.In this direction much work followed subsequently, for example, [14][15][16][17][18][19][20][21][22][23].
Though Pu and Liu [24] defined the notion of fuzzy boundary in fuzzy topological spaces in 1980, yet there is very little work available on this notion in present literature.One reason, inter alia, of Tang's [8] use of a limited version of Chang's fuzzy topological space was the nonavailability of sufficient material about properties of fuzzy boundary.So, we study this concept further and establish several of its properties, thus providing sufficient material for researchers to utilize these concepts fruitfully.Ahmad and Athar [25] defined the concept of fuzzy semiboundary and characterized fuzzy semicontinuous functions in terms of fuzzy semiboundary.
In this paper, we present several properties of fuzzy boundary and fuzzy semiboundary which have been supported by examples.Properties of fuzzy semi-interior, fuzzy semiclosure, fuzzy boundary, and fuzzy semiboundary have been obtained in product-related spaces.We give necessary conditions for fuzzy continuous (resp., fuzzy semicontinuous, fuzzy irresolute) functions.Moreover, fuzzy continuous Advances in Fuzzy Systems (resp., fuzzy semicontinuous, fuzzy irresolute) functions have been characterized via fuzzy-derived (resp., fuzzy semiderived) sets.

PRELIMINARIES
First, we briefly recall certain definitions and results; for those not described; we refer to [1,2,13,22].
A fuzzy set λ in a set X is a function from Definition 1 (see [1]).Let λ and μ be fuzzy sets in X.Then, for all x ∈ X, ( More generally, for a family Λ = {λ i | i ∈ I} of fuzzy sets in X, the union ψ = ∨ i λ i and intersection δ = ∧ i λ i are defined by ( The empty fuzzy set 0 is defined as 0(x) = 0, for all x ∈ X, and the symbol 1 denotes the fuzzy set 1(x) = 1, for all x ∈ X. Definition 2 (see [2]).Let f : X → Y be a function.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 , where Definition 3 (see [2]).A fuzzy topology is a family τ of fuzzy sets in X, which satisfies the following conditions: (1) 0, 1 ∈ τ; τ is called a fuzzy topology for X, and the pair (X, τ) is a fuzzy topological space.Every member of τ is called τ-open

fuzzy set (or simply fuzzy open 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 0 and 1, while the discrete fuzzy topology contains all fuzzy sets.In the sequel, we write an fts X (or (X, τ)) in place of "a space X with fuzzy topology τ." Definition 4 (see [2]).The closure and interior of a fuzzy set λ in an fts (X, τ) are denoted and defined as We mention below some properties of closure and interior of a fuzzy set which will be used in the sequel.
Lemma 1 (see [26]).For fuzzy sets λ and μ in an fts X, one has the following: Definition 5 (see [2]).A function f : (X, τ) → (Y , δ) is said to be fuzzy continuous if and only if the inverse of each δfuzzy open set is τ-fuzzy open.

FUZZY BOUNDARY
Definition 6 (see [24]).Let λ be a fuzzy set in an fts X.Then, the fuzzy boundary of λ is defined as Bd λ = Cl λ ∧ Cl λ c .Obviously, Bd λ is a fuzzy closed set.
Remark 1.In classical topology, for an arbitrary set A of a topological space X, we have A ∪ Bd A = Cl A, but in fuzzy topology we have λ ∨ Bd λ ≤ Cl λ, for an arbitrary fuzzy set λ in X, the converse of which is not true as shown by Pu and Liu [24].Moreover, we have the following proposition.
Proposition 1.For fuzzy sets λ and μ in an fts X, the following conditions hold.
The following proposition gives some more properties of fuzzy boundary.

Proposition 2.
Let λ be a fuzzy set in an fts X.Then, The following example shows that the equality does not hold in Proposition 2(2)-( 4).
Remark 2. In general topology, the following conditions hold: whereas, in fuzzy topology, we give counter-examples to show that these may not hold in general.
Theorem 2. Let λ and μ be fuzzy sets in an fts X.Then, Bd (λ∨ Proof.
In Theorem 2, the equality does not hold as is shown by the following.

Advances in Fuzzy Systems
The following examples show that For this, choose However, we have the following theorem.
Theorem 3.For any fuzzy sets λ and μ in an fts X, one has Proof.
Corollary 1.For any fuzzy sets λ and μ in an fts X, one has Example 5. To show that the equality in Theorem 3, in general, does not hold, choose λ = {a .6 , b .6 , c .7 } and μ = {a .6 , b .2, c .9 } in the fts X defined in Example 1.Then, calculations give In general topology, it is known that for any subset A of a space X.However, in fuzzy topology, we have the following proposition.
Proposition 3.For any fuzzy set λ in an fts X, one has Remark 3. We could not find an example to show that the equality in (2) does not hold .However, the equality in (1), in general, does not hold as is shown by the following example.
Definition 7 (see [27]).If λ is a fuzzy set of X and μ is a fuzzy set of Y , then Definition 8 (see [13]).An fts (X, τ X ) is product related to another fts (Y , τ Y ) if for any fuzzy set ν of X and ξ of Y whenever λ c / ≥ ν and Theorem 4 (see [13]).Let X and Y be product-related fts's.
Then, for a fuzzy set λ of X and a fuzzy set μ of Y , one has Lemma 2. For fuzzy sets λ, μ, ν, and ω in a set X, one has Proof.
It suffices to prove this for n = 2. Consider Bd Theorem 6.Let f : X → Y be a fuzzy continuous function.Then, for any fuzzy set μ in Y .
Proof.Let f be fuzzy continuous and μ a fuzzy set in Y .Then, Therefore, Bd Before closing this section, we give an interesting characterization of fuzzy continuous functions in terms of fuzzyderived set and fuzzy closure.It shows that fuzzy continuity, in essence, amounts to preservation of fuzzy closedness.For this we first recall following definitions.
Definition 9 (see [24]).A fuzzy set in X is called a fuzzy point if and only if it takes the value 0 for all x ∈ X except one, say, e ∈ X.If its value at e is α(0 < α ≤ 1), one denotes this fuzzy point by e α .The point e ∈ X is called support of fuzzy point e α and is denoted as supp (e).
Definition 11 (see [24]).A fuzzy set λ in an fts X is called a Q-neighborhood of a fuzzy point e, if there exists a μ ∈ τ such that eqμ ≤ λ.
Definition 12 (see [24]).A fuzzy point e is called an adherent point of a fuzzy set λ, if every Q-neighborhood of e is quasicoincident with λ.
Definition 13 (see [24]).A fuzzy point e is called an accumulation point of a fuzzy set λ, if e is an adherent point of λ and every Q-neighborhood of e and λ is quasicoincident at some point different from supp (e), whenever e ∈ λ.The union of all the accumulation points of λ is called the derived set of λ, denoted as λ d .It is evident that λ d ≤ Cl λ.Proposition 4 (see [24]).Let λ be a fuzzy set in an fts X, then (2) λ is fuzzy closed if and only if λ d ≤ λ.
We use Proposition 4 and prove the following theorem.
Theorem 7. Let f : X → Y be a function.Then, the following conditions are equivalent: (1) f : X → Y is fuzzy continuous; (2) f (λ d ) ≤ Cl f (λ), for any fuzzy set λ in X.
Proof.(1)⇒(2) Let f be fuzzy continuous and λ a fuzzy set in X.
Thus, f is fuzzy continuous.
Definition 15 (see [22]).Let λ be a fuzzy set in an fts X.Then, semiclosure (briefly sCl) and semi-interior (briefly sInt) of λ are given as Remark 4. In the following theorems, we note that almost all the properties related to fuzzy semi-interior, fuzzy semiclosure, and fuzzy semiboundary are analogous to their counterparts in fuzzy topology and hence some of the proofs are not given.
The inequalities ( 1) and ( 4) of Theorem 8 are irreversible as is shown by the following example.
Remark 5.In fuzzy topology, we have λ ∨ s Bd λ ≤ sCl λ, for an arbitrary fuzzy set λ in X, the equality does not hold as the following example shows.
In the following theorem, (1)-( 5) are analogs of Proposition 1 and hence we omit their proofs.Proposition 5.For a fuzzy set λ in an fts X, the following conditions hold.
Proof.(6) Since sCl λ ≤ Cl λ and sCl λ c ≤ Cl λ c , then we have The converse of ( 2) and ( 3) and reverse inequalities of ( 6) and ( 7) in Proposition 5 are, in general, not true as is shown by the following example.
The following is analog of Proposition 2 and hence we omit its proof.Proposition 6.Let λ be a fuzzy set in an fts X.Then, one has To show that the inequalities (2), (3), and (4) of Proposition 6 are, in general, irreversible, we have the following example.(37) Remark 6.In general topology, the following conditions hold: whereas, in fuzzy topology, we give counter-examples to show that these may not hold in general.
Example 11.In the fts X of Example 8, we choose fuzzy set λ = {a .7 , b .5 }, then calculations give It is easily seen that sInt λ ∨ sBd λ ≤ sCl λ.
The reverse inequality in Theorem 9 is, in general, not true as is shown by the following example.The following example shows that For this choose γ = {a .4, b .8} and δ = {a .6 , b .3}.Then, calculations give However, we have the following theorem which is an analog of Theorem 3.
Theorem 10.For any fuzzy sets λ and μ in an fts X, one has Corollary 2. For any fuzzy sets λ and μ in an fts X, one has Example 13.To show that the reverse inequality in Theorem 10 is, in general, not true, choose fuzzy sets γ and δ as given in Example 8.Then, calculations give The analog of Proposition 3 is the following theorem, the proof of which is similar.Proposition 7.For any fuzzy set λ in an fts X, one has (1) sBd sBd λ ≤ sBd λ, (2) sBd sBd sBd λ ≤ sBd sBd λ.
Remark 7. As in the case of Proposition 3(2), we also do not know whether the equality in Proposition 7(2) holds or not.However, the reverse inequality of (1) is, in general, not true as is shown by the following example.Lemma 3 (see [13]).If λ is a fuzzy set of X and μ is a fuzzy set of Y , then Using Lemma 3, we have the following one.
Using Lemma 4, we have the following theorem.
Theorem 11.If λ is a fuzzy set of fts X and μ of fts Y , then Moreover, we have the following one.
Theorem 12. Let X and Y be product-related fts's.Then, for a fuzzy set λ of X and a fuzzy set μ of Y , one has Proof.(1) For fuzzy sets λ i s of X and μ j s of Y , we first note that In view of Theorem 11, it is sufficient to show that sCl(λ × μ) ≥ sCl λ × sCl μ.Let λ i ∈ FSO(X) and μ j ∈ FSO(Y ).Then, we have (2) This follows from (1) using the facts that (sInt ψ) c = sCl ψ c and (sCl ψ) c = sInt ψ c .The analog of Theorem 5 is the following, the proof of which is similar.Theorem 13.Let X i , i = 1, 2, . . ., n, be a family of productrelated fuzzy topological spaces.If each λ i is a fuzzy set in X i , then The following theorem gives a necessary condition for fuzzy semicontinuous functions in terms of fuzzy boundary and fuzzy semiboundary.Theorem 14.Let f : X → Y be a fuzzy semicontinuous function.Then, one has for any fuzzy set μ in Y .
Proof.Let f be fuzzy semicontinuous and μ a fuzzy set in Y .Then, Cl μ is fuzzy closed in Y , which implies that f −1 (Cl μ) is fuzzy semiclosed in X.Therefore, Definition 17 (see [28]).A function f : X → Y is said to be fuzzy irresolute if f −1 (β) is fuzzy semiopen in X, for each fuzzy semiopen set β in Y .
The following theorem gives a necessary condition of fuzzy irresolute functions in terms of fuzzy boundary and fuzzy semiboundary, the proof of which is similar to Theorem 14.
Theorem 15.Let f : X → Y be a fuzzy irresolute function.Then, one has Definition 21 (see [13]).Let f : X → Y be a function from an fts X to another fts Y.Then, f is said to be fuzzy semicontinuous function if f −1 (λ) is fuzzy semiopen in X, for each fuzzy open set λ in Y.
We use Corollary 3 and characterize fuzzy semicontinuous functions in terms of fuzzy semiderived set as follows.
Theorem 17.Let f : X → Y be a function.Then, the following conditions are equivalent: (1) f is fuzzy semicontinuous, (2) f (λ sd ) ≤ Cl f (λ), for any fuzzy set λ in X.
Finally, we characterize fuzzy irresolute functions via fuzzy semiderived set as follows.

)
for any fuzzy set μ in Y .Definition 18.A fuzzy set λ in an fts X is called a fuzzy semi-Q-neighborhood of a fuzzy point e, if there exists a fuzzy semiopen set μ in X, such that eqμ ≤ λ.Theorem 16.A fuzzy point e= x α ∈ sCl λ if each semi-Qneighborhood of e is quasicoincident with λ.Proof.xα∈ sCl λ if and only if for every fuzzy closed set ψ ≥ λ, x α ∈ ψ.This gives ψ(x α ) ≥ λ(x α ).Equivalently, x α ∈ sCl λ if and only if for every fuzzy semiopen setβ ≤ λ c , β(x α ) ≤ λ c (x α ).That is, for every fuzzy open set β satisfying β(x α ) ≥ λ c , β is not contained in λ c , or βqλ cc = λ.Thus, x α ∈ sCl λ if every fuzzy open Q-neighborhood β of x α is quasicoincident with λ.Definition 19.A fuzzy point e is called semiadherent point of a fuzzy set λ if every semi-Q-neighborhood of e is quasicoincident with λ.Definition 20.A fuzzy point e is called a semiaccumulation point of a fuzzy set λ if e is a semiadherent point of λ and every semi-Q-neighborhood of e and λ is quasicoincident at some point different from supp (e), whenever e ∈ λ.The union of all the semiaccumulation points of λ is called the fuzzy semiderived set of λ, denoted as λ sd .It is evident that λ sd ≤ sCl λ.Let λ be a fuzzy set in X, then sCl λ = λ ∨ λ sd .Proof.Let Ω = {e | e is a semiadherent point of λ}.Then, from Theorem 16, sCl λ = ∨Ω.On the other hand, e ∈ Ω is either e ∈ λ or e / ∈ λ; for the latter case, by Definition 20, e ∈ λ sd , hence sCl λ = ∨Ω ≤ λ ∨ λ sd .The reverse inclusion is obvious.For any fuzzy set λ in an fts X, λ is fuzzy semiclosed if and only if λ sd ≤ λ.