# Fuzzy Boundary and Fuzzy Semiboundary

Authors: M. Athar; B. Ahmad
Journal: Advances in Fuzzy Systems
 (2008)

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## Abstract

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.


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## Body

## 1. 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 T1/2-space [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,
[14–23].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 (resp., fuzzy semicontinuous, fuzzy
irresolute) functions have been characterized via fuzzy-derived (resp., fuzzy
semiderived) sets.


## 2. Preliminaries

First, we briefly recall certain definitions and
results; for those not described; we refer to [1, 2, 13, 22].A fuzzy setλ in a set X is a function from X to [0,1],
that is, λ:X→[0,1].Definition 1 (see [1]).
Letλ and μ be fuzzy sets in X.
Then, for all x∈X,
(1)λ=μλ(x)=μ(x),λ≤μλ(x)≤μ(x),ψ=λ∨μψ(x)=max⁡{λ(x),μ(x)},δ=λ∧μδ(x)=min⁡{λ(x),μ(x)},η=λcη(x)=1−λ(x).More generally, for a familyΛ={λi∣i∈I} of fuzzy sets in X, the union ψ=∨iλi and intersection δ=∧iλi are defined by(2)ψ(x)=supi{λi(x)∣x∈X},δ(x)=inf⁡i{δi(x)∣x∈X}.The empty fuzzy set0 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]).
Letf: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(3)f−1(β)(x)=β(f(x)),∀x∈X.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(4)f(λ)(y)={supx∈f−1(y){λ(x)},iff−1(y)isnonempty,0,otherwise,for all y∈Y,
where f−1(y)={x∣f(x)=y}.Definition 3 (see [2]).
A fuzzy topology is a familyτ of fuzzy sets in X,
which satisfies the following conditions:
(1)
0,1∈τ;(2)
ifλ,μ∈τ,
then λ∧μ∈τ;(3)
ifλi∈τ for each i∈I,
then ∨iλi∈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 only0 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 setλ in an fts (X,τ) are denoted and defined as(5)Clλ=inf{μ∣λ≤μ,μc∈τ},Intλ=sup{ω∣ω≤λ,ω∈τ}.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 setsλ and μ in an fts X, one has the
following:
(1)
λ is fuzzy closed (resp., fuzzy open) ⇔Clλ=λ (resp., Intλ=λ);(2)
λ≤μ⇒Clλ≤Clμ (Intλ≤Intμ);(3)
ClClλ=Clλ (IntIntλ=Intλ);(4)
Clλ∨Clμ=Cl(λ∨μ);(5)
Clλ∧Clμ≥Cl(λ∧μ);(6)
Intλ∨Intμ≤Int(λ∨μ);(7)
Intλ∧Intμ=Int(λ∧μ);(8)
Intλc=(Clλ)c;(9)
Clλc=(Intλ)c.Definition 5 (see [2]).
A functionf:(X,τ)→(Y,δ) is said to be fuzzy continuous if and only if
the inverse of each δ-fuzzy open set is τ-fuzzy open.Theorem 1 (see [19, 26]).
Letf:X→Y be a function from an fts (X,τ) to another fts (Y,δ). Then, the following conditions are
equivalent:
(1)
f is fuzzy continuous,(2)
the inverse of eachδ-closed fuzzy set is τ-closed,(3)
for each fuzzy setλ in X,  f(Clλ)≤Clf(λ),(4)
for each fuzzy setλ in Y,  f−1(Intν)≤Intf−1(ν).


## 3. Fuzzy Boundary

Definition 6 (see [24]).
Letλ be a fuzzy set in an fts X. Then, the fuzzy boundary of λ is defined as Bdλ=Clλ∧Clλc. Obviously, Bdλ is a fuzzy closed set.Remark 1.
In classical topology, for an arbitrary setA of a topological space X,
we have A∪BdA=ClA,
but in fuzzy topology we have λ∨Bdλ≤Clλ,
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 setsλ and μ in an fts X, the following
conditions hold.
(1)
Bdλ=Bdλc.(2)
Ifλ is fuzzy closed, then Bdλ≤λ.(3)
Ifλ is fuzzy open, then Bdλ≤λc.(4)
Letλ≤μ and μ∈FC(X) (resp., μ∈FO(X)) . Then, Bdλ≤μ (resp., Bdλ≤μc), where FC(X) (resp., FO(X)) denotes the class of fuzzy closed (resp.,
fuzzy open) sets in X.(5)
(Bdλ)c=Intλ∨Intλc.Proof.
(1)Bdλ=Clλ∧Clλc=Clλc∧Clλ=Clλc∧Cl(λc)c=Bdλc.
(2)Bdλ=Clλ∧Clλc≤Clλ=λ, hence Bdλ≤λ.
(3)λ is fuzzy open implies λc is fuzzy closed. By (2),  Bdλc≤λc and by (1), we get Bdλ≤λc.
(4) Sinceλ≤μ implies Clλ≤Clμ, we have Bdλ=Clλ∧Clλc≤Clμ∧Clλc≤Clμ=μ, since μ∈FC(X).
(5)(Bdλ)c=(Clλ∧Clλc)c=(Clλ)c∨(Clλc)c=Intλc∨Intλ.The converse of (2) and (3) of Proposition1 is, in general, not true as is
shown by the following example.Example 1.
LetX={a,b,c} be a set and τ,
the fuzzy topology generated by fuzzy sets μ={a.4,b.8,c.2},ν={a.6,b.9,c.1}, and ω={a.5,b.7,c.3}. Then, τ={0,μ,ν,ω,{a.4,b.8,c.1},{a.4,b.7,c.2},{a.5,b.7,c.1},{a.4,b.7,c.1},{a.5,b.8,c.3},{a.5,b.8,c.1},{a.6,b.9,c.2},{a.5,b.8,c.2},{a.6,b.9,c.3},{a.5,b.7,c.2},1}. Choose λ={a.5,b.7,c.9} and ψ={a.5,b.3,c.3}. Calculations give(6)Bdλ={a.5,b.3,c.7}≤λ,butλisnotfuzzyclosed,Bdψ={a.5,b.3,c.7}≤ψc,butψisnotfuzzyopen.The following proposition gives some more properties of fuzzy boundary.Proposition 2.
Letλ be a fuzzy set in an fts X.
Then,
(1)
Bdλ=Clλ−Intλ;(2)
BdIntλ≤Bdλ;(3)
BdClλ≤Bdλ;(4)
Intλ≤λ−Bdλ.Proof.
(1) Since(Clλc)c=Intλ, therefore we have Bdλ=Clλ∧Clλc=Clλ−(Clλc)c=Clλ−Intλ. Thus Bdλ=Clλ−Intλ. This proves (1),
(2)BdIntλ=ClIntλ−IntIntλ=ClIntλ−Intλ≤Clλ−Intλ=Bdλ,
(3)BdClλ=ClClλ−IntClλ=Clλ−IntClλ≤Clλ−Intλ=Bdλ,
(4)(7)λ−Bdλ=λ∧(Bdλ)c=λ∧(Clλ∧Clλc)c=λ∧(Intλc∨Intλ)=(λ∧Intλc)∨(λ∧Intλ)=(λ∧Intλc)∨Intλ≥Intλ.The following example shows that the equality does not hold in Proposition2(2)–(4).Example 2.
Chooseα={a.1,b.4,c.7},β={a.6,b.8,c.5}, and γ={a.6,b.3,c.8} in Example 1. Then, calculations give(8)Bdα=1≠0=BdIntα,Bdβ={a.5,b.2,c.7}≠0=BdClβ,γ−Bdγ={a.4,b.3,c.2}≠Intγ=0.Remark 2.
In general topology, the following conditions
hold:(9)BdA∩IntA=ϕ,IntA∪BdA=ClA,IntA∪IntAc∪BdA=X,whereas, in fuzzy topology, we
give counter-examples to show that these may not hold in general.Example 3.
In the fts(X,τ) of Example 1, we choose fuzzy set λ={a.4,b.7,c.7}, then calculations give(10)Bdλ∧Intλ={a.6,b.3,c.8}∧{a.4,b.7,c.2}≠0,Intλ∨Bdλ={a.6,b.7,c.8}≠Clλ=1,Intλ∨Intλc∨Bdλ={a.6,b.7,c.8}≠1.It is easily seen that Intλ∨Bdλ≤Clλ.Theorem 2.
Letλ and μ be fuzzy sets in an fts X. Then, Bd(λ∨μ)≤Bdλ∨Bdμ.Proof.
(11)Bd(λ∨μ)=Cl(λ∨μ)∧Cl(λ∨μ)c≤(Clλ∨Clμ)∧(Clλc∧Clμc)=[Clλ∧(Clλc∧Clμc)]∨[Clμ∧(Clλc∧Clμc)]=(Bdλ∧Clμc)∨(Bdμ∧Clλc)≤Bdλ∨Bdμ.In Theorem2, the equality does not hold as is shown by the following:Example 4.
In Example1, choose fuzzy sets α={a0,b0,c.2} and β={a.4,b.8,c.1}. Then, calculations give(12)Bdα∨Bdβ={a.6,b.2,c.9}≠{a.6,b.2,c.8}=Bd(α∨β).The following examples show
that(13)Bd(γ∧δ)≰Bdγ∧Bdδ,Bdψ∧Bdξ≰Bd(ψ∧ξ).For this, choose γ={a.4,b.1,c.9},δ={a.4,b1,c1},ψ={a.7,b.4,c.4},
and ξ={a.4,b.1,c.8}. Then,  calculations give(14)Bd(γ∧δ)={a.4,b.1,c.9}≰{a.4,b.1,c.8}=Bdγ∧Bdδ,Bdψ∧Bdξ={a.4,b.1,c.8}≰{a.4,b.1,c.7}=Bd(ψ∧ξ).However, we have the following
theorem.Theorem 3.
For any fuzzy setsλ and μ in an fts X, one
has(15)Bd(λ∧μ)≤(Bdλ∧Clμ)∨(Bdμ∧Clλ).Proof.
(16)Bd(λ∧μ)=Cl(λ∧μ)∧Cl(λ∧μ)c≤(Clλ∧Clμ)∧(Clλc∨Clμc)=[(Clλ∧Clμ)∧Clλc]∨[(Clλ∧Clμ)∧Clμc]=(Bdλ∧Clμ)∨(Bdμ∧Clλ).Corollary 1.
For any fuzzy setsλ and μ in an fts X, one
has(17)Bd(λ∧μ)≤Bdλ∨Bdμ.Example 5.
To show
that the equality in Theorem3, 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(18)(Bdλ∧Clμ)∨(Bdμ∧Clλ)=μ≰{a.6,b.2,c.8}=Bd(λ∧μ).In general topology, it is known that(19)BdBdBdA=BdBdA,for any subset A of a space X. However, in fuzzy topology, we have the
following proposition.Proposition 3.
For any fuzzy
setλ in an fts X, one has
(1)
BdBdλ≤Bdλ;(2)
BdBdBdλ≤BdBdλ.Proof.
(1)BdBdλ=ClBdλ∧Cl(Bdλ)c≤ClBdλ=Bdλ.
(2)(20)BdBdBdλ=ClBdBdλ∧Cl(BdBdλ)c=BdBdλ∧Cl(BdBdλ)c≤BdBdλ.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.Example 6.
Chooseλ={a.2,b.5,c.9} in fts X of Example 1. Then, Bdλ=1 but BdBdλ=0.Definition 7 (see [27]).
Ifλ is a fuzzy set of X and μ is a fuzzy set of Y, then(21)(λ×μ)(x,y)=min⁡{λ(x),μ(y)},foreach(x,y)∈X×Y.Definition 8 (see [13]).
An fts(X,τX) is product related to another fts (Y,τY) if for any fuzzy set ν of X and ξ of Y whenever λc≱ν and μc≱ξ imply λc×1∨1×μc≥ν×ξ, where λ∈τX and μ∈τY, there exist λ1∈τX and μ1∈τY such that λ1c≥ν and μ1c≥ξ and λ1c×1∨1×μ1c=λc×1∨1×μc.Theorem 4 (see [13]).
LetX and Y be product-related fts's. Then, for a fuzzy
set λ of X and a fuzzy set μ of Y, one has
(1)
Cl(λ×μ)=Clλ×Clμ,(2)
Int(λ×μ)=Intλ×Intμ.Lemma 2.
For fuzzy
setsλ, μ, ν,
and ω in a set X, one
has(22)(λ∧μ)×(ν∧ω)=(λ×ω)∧(μ×ν).Proof.
(23)((λ∧μ)×(ν∧ω))(x,y)=min⁡((λ∧μ)(x),(μ∧ω)(y))=min⁡(min⁡(λ(x),μ(x)),min⁡(ν(y),ω(y)))=min⁡(min⁡(λ(x),ω(y)),min⁡(μ(x),ν(y)))=min⁡((λ×ω)(x,y),(μ×ν)(x,y))=((λ×ω)∧(μ×ω))(x,y).Theorem 5.
LetXi,i=1,2,…,n, be a family of product-related fuzzy
topological spaces. If each λi is a fuzzy set in Xi, then(24)Bd∏i=1nλi=[Bdλ1×Clλ2×⋯×Clλn]∨[Clλ1×Bdλ2×Clλ3×⋯×Clλn]∨⋯∨[Clλ1×Clλ2×⋯×Bdλn].Proof.
We use Proposition2(1),
Theorem 4, and Lemma 2 to prove this.
It suffices to prove this forn=2. Consider
(25)Bd(λ1×λ2)=Cl(λ1×λ2)−Int(λ1×λ2)=(Clλ1×Clλ2)−(Intλ1×Intλ2)=(Clλ1×Clλ2)−((Intλ1∧Clλ1)×(Intλ2∧Clλ2))=(Clλ1×Clλ2)−((Intλ1×Clλ2)∧(Clλ1×Intλ2))(byLemma2)=[(Clλ1×Clλ2)−(Intλ1×Clλ2)]∨[(Clλ1×Clλ2)−(Clλ1×Intλ2)]=[(Clλ1−Intλ1)×Clλ2]∨[Clλ1×(Clλ2−Intλ2)]=(Bdλ1×Clλ2)∨(Clλ1×Bdλ2).Theorem 6.
Letf:X→Y be a fuzzy continuous function.
Then,(26)Bdf−1(μ)≤f−1(Bdμ),for any fuzzy set μ in Y.Proof.
Letf be fuzzy continuous and μ a fuzzy set in Y.
Then,
(27)Bdf−1(μ)=Clf−1(μ)∧Cl(f−1(μ))c≤Clf−1(Clμ)∧Clf−1(Clμc)=f−1(Clμ)∧f−1(Clμc)=f−1(Clμ∧Clμc)=f−1(Bdμ).Therefore, Bdf−1(μ)≤f−1(Bdμ).Before closing this section, we give an interesting
characterization of fuzzy continuous functions in terms of fuzzy-derived 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 inX 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 eα. The point e∈X is called support of fuzzy point eα and is denoted as supp(e).Definition 10 (see [24]).
A fuzzy pointeα is said to be quasicoincident (also written as
q-coincident or Q-coincident) with a fuzzy set λ, denoted by eαqλ, if for some x∈X,α>λc(x),
or α+λ(x)>1.Definition 11 (see [24]).
A fuzzy setλ 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 pointe 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 pointe 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 λd. It is evident that λd≤Clλ.Proposition 4 (see [24]).
Letλ be a fuzzy set in an fts X,
then
(1)
Clλ=λ∨λd;(2)
λ is fuzzy closed if and only if λd≤λ.We use Proposition4 and prove the following theorem.Theorem 7.
Letf:X→Y be a function. Then, the following
conditions are equivalent:
(1)
f:X→Y is fuzzy continuous;(2)
f(λd)≤Clf(λ), for any fuzzy set λ in X.Proof.
(1)⇒(2) Let f be fuzzy continuous and λ a fuzzy set in X.
Since Clf(λ) is fuzzy closed in Y, then f−1(Clf(λ)) is fuzzy closed in X. λ≤f−1(Clf(λ)) gives Clλ≤Clf−1(Clf(λ))=f−1(Clf(λ)). Therefore, f(λd)≤f(Clλ)≤ff−1(Clf(λ))≤Clf(λ). Consequently, f(λd)≤Clf(λ).
(2)⇒(1) Suppose f(λd)≤Clf(λ),λ∈IX.
Let μ be any fuzzy closed set in Y. We show that f−1(μ) is fuzzy closed in X. By our hypothesis, f([f−1(μ)]d)≤Clf(f−1(μ))≤Clμ=μ,f([f−1(μ)]d)≤μ gives [f−1(μ)]d≤f−1(f[f−1(μ)]d)≤f−1(μ), or [f−1(μ)]d≤f−1(μ) implies that f−1(μ) is fuzzy closed in X. Thus, f is fuzzy continuous.


## 4. Fuzzy Semiboundary

First, we recall some notions.Definition 14 (see [13]).
Letλ be a fuzzy set in an fts (X,τ).
Then, λ is called a fuzzy semiopen set of X if there exists a ν∈τ such that ν≤λ≤Clν.Definition 15 (see [22]).
Letλ be a fuzzy set in an fts X.
Then, semiclosure (briefly sCl) and semi-interior (briefly sInt) of λ are given as(28)sClλ=∧{β∣λ≤β,βfuzzysemiclosed},sIntλ=∨{β∣β≤λ,βfuzzysemiopen}.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.Theorem 8.
For fuzzy setsλ and μ in an fts X,
one has
(1)
sInt(λ∨μ)≥sIntλ∨sIntμ;(2)
sInt(λ∧μ)=sIntλ∧sIntμ;(3)
sCl(λ∨μ)=sClλ∨sClμ;(4)
sCl(λ∧μ)≤sClλ∧sClμ;(5)
sClsClλ=sClλ;(6)
sIntsIntλ=sIntλ;(7)
(sIntλ)c=sCl(λc);(8)
(sClλ)c=sInt(λc).Proof.
(1)sIntλ and sIntμ are both fuzzy semiopen and λ≤λ∨μ, μ≤λ∨μ imply sIntλ≤sInt(λ∨μ) and sIntμ≤sInt(λ∨μ).
Combining, sIntλ∨sIntμ≤sInt(λ∨μ) or(29)sInt(λ∨μ)≥sIntλ∨sIntμ.
(2)λ∧μ≤λ and λ∧μ≤μ imply sInt(λ∧μ)≤sIntλ,sInt(λ∧μ)≤sInttμ and therefore sInt(λ∧μ)≤sIntλ∧sIntμ. Conversely sIntλ≤λ and sIntμ≤μ implies sIntλ∧sIntμ≤λ∧μ and sIntλ∧sIntμ is fuzzy semiopen. But sInt(λ∧μ) is the largest fuzzy semiopen set contained in λ∧μ, hence sInt∧sIntμ≤sInt(λ∧μ). This gives the equality.
(3) It follows easily from (2).
(4) Sinceλ∧μ≤λ,λ∧μ≤μ(30)⟹sCl(λ∧μ)≤sClλ,sCl(λ∧μ)≤sClμ⟹sCl(λ∧μ)≤sClλ∧sClμ.
(5)–(8) Proofs are straightforward.The inequalities (1) and (4) of Theorem8 are irreversible as is shown by
the following example.Example 7.
LetX={a,b} be a set and I={0,.5,1} the lattice of membership grades for fuzzy
sets in X.
Let τ be the fuzzy topology on X given as(31)τ={0,{a.5,b1},{a.0,b1},{a1,b.5},{a.5,b.5},{a0,b.5},1}.We choose fuzzy sets α={a.5,b0},β={a0,b.5},γ={a.5,b1},
and δ={a1,b.5}. Then, calculations give(32)sInt(α∨β)={a.5,b.5}≰{a0,b.5}=sIntα∨sIntβ,sClγ∧sClδ={a1,b.5}≰{a.5,b.5}=sCl(α∧δ).Definition 16 (see [25]).
Letλ be a fuzzy set in an fts X. Then, the fuzzy semiboundary of λ is defined as sBdλ=sClλ∧sClλc. Obviously, sBdλ is a fuzzy semiclosed set.Remark 5.
In fuzzy topology, we haveλ∨sBdλ≤sClλ,
for an arbitrary fuzzy set λ in X,
the equality does not hold as the following example shows.Example 8.
LetX={a,b} be a set with fuzzy topology τ={0,{a.4,b.8},{a.6,b.9},{a.5,b.7},{a.5,b.2},{a.8,b.7},{a.3,b.2},{a.4,b.7},{a.4,b.2},{a.6,b.7},{a.5,b.8},{a.8,b.8},{a.6,b.8},{a.8,b.9},1}. Choose λ={a.4,b.7}, then calculations give(33)sClλ={a.5,b.8}≠{a.5,b.7}=λ∨sBdλ.In the following theorem, (1)–(5) are analogs of
Proposition1 and hence we omit their proofs.Proposition 5.
For a fuzzy setλ in an fts X, the following
conditions hold.
(1)
sBdλ=sBdλc.(2)
Ifλ is fuzzy semiclosed, then sBdλ≤λ.(3)
Ifλ is fuzzy semiopen, then sBdλ≤λc.(4)
Letλ≤μ and μ∈FSC(X) (resp., μ∈FSO(X)) . Then, sBdλ≤μ (resp., sBdλ≤μc), where FSC(X) (resp., FSO(X)) denotes the class of fuzzy semiclosed
(resp., fuzzy semiopen) sets in X.(5)
(sBdλ)c=sIntλ∨sIntλc.(6)
sBdλ≤Bdλ.(7)
sClsBdλ≤Bdλ.Proof.
(6) SincesClλ≤Clλ and sClλc≤Clλc,
then we have(34)sBdλ=sClλ∧sClλc≤Clλ∧Clλc=Bdλ.
(7)(35)sClsBdλ=sCl(sClλ∧sClλc)≤sClsClλ∧sClsClλc=sClλ∧sClλc=sBdλ≤Bdλ.The converse of (2) and (3) and reverse inequalities of (6) and (7) in Proposition5 are, in general, not true as is
shown by the following example.Example 9.
Chooseλ={a.7,b.7}, ψ={a.3,b.3},ω={a.7,b.9}, ξ={a.6,b1} in the fts (X,τ) defined in Example 8. Then, calculations
give(36)sBdλ={a.4,b.3}≤λ,butλisnotfuzzysemiclosed,sBdψ={a.4,b.3}≤ψc,butψisnotfuzzysemiopen,Bdω={a.4,b.1}≰{a.3,b.1}=sBdω,Bdξ={a.4,b.1}≰{a.4,b0}=sClsBdξ.The following is analog of Proposition2 and hence we omit its proof.Proposition 6.
Letλ be a fuzzy set in an fts X.
Then, one has
(1)
sBdλ=sClλ−sIntλ,(2)
sBdsIntλ≤sBdλ,(3)
sBdsClλ≤sBdλ,(4)
sIntλ≤λ−sBdλ.To show that the inequalities (2), (3),
and (4) of Proposition6, are, in general, irreversible,
we have the following example.Example 10.
Chooseα={a.7,b.5} and μ={a.6,b.3} in the fts X defined in Example 8. Then, calculations
give(37)sBdα={a.5,b.8}≰{a.5,b.2}=sBdsIntα,sBdα={a.5,b.8}≰{a.4,b.2}=sBdsClα,μ−sBdμ={a.5,b.3}≰{a.5,b.2}=sIntμ.Remark 6.
In general topology, the following conditions
hold:(38)sBdA∩sIntA=ϕ,sIntA∪sBdA=ClA,sIntA∪sIntAc∪sBdA=X,whereas, in fuzzy topology, we
give counter-examples to show that these may not hold in general.Example 11.
In the ftsX of Example 8, we choose fuzzy set λ={a.7,b.5}, then calculations give(39)sBdλ∧sIntλ={a.5,b.8}∧{a.5,b.2}≠0,sIntλ∨sBdλ={a.5,b.8}≠{a.7,b.8}=sClλ,sIntλ∨sIntλc∨sBdλ={a.5,b.8}≠1.It is easily seen that sIntλ∨sBdλ≤sClλ.Theorem 9.
Letλ and μ be fuzzy sets in an fts X. Then, sBd(λ∨μ)≤sBdλ∨sBdμ.The reverse inequality in Theorem9 is, in general,
not true as is shown by the following example.Example 12.
In Example8, choose fuzzy sets α={a.4,b.7} and β={a.6,b.2}. Then, calculations give(40)sBdα∨sBdβ={a.5,b.3}≰{a.4,b.3}=sBd(α∨β).The following example shows
that(41)sBd(γ∧δ)≰sBdγ∧sBdδ,sBdλ∧sBdδ≰sBd(γ∧δ).For this choose γ={a.4,b.8} and δ={a.6,b.3}. Then, calculations give(42)sBd(γ∧δ)={a.4,b.3}≰{a.5,b.2}=sBdγ∧sBdδ,sBdλ∧sBdδ={a.5,b.2}≰{a.4,b.3}=sBd(γ∧δ),However, we have the following
theorem which is an analog of Theorem 3.Theorem 10.
For any fuzzy setsλ and μ in an fts X, one
has(43)sBd(λ∧μ)≤(sBdλ∧sClμ)∨(sBdμ∧sClλ).Corollary 2.
For any fuzzy setsλ and μ in an fts X, one has(44)sBd(λ∧μ)≤sBdλ∨sBdμ.Example 13.
To show that the reverse inequality in Theorem10 is, in general, not true, choose
fuzzy sets γ and δ as given in Example 8. Then, calculations
give(45)(sBdγ∧sClδ)∨(sBdγ∧sClδ)={a.5,b.3}≰{a.4,b.3}=sBd(γ∧δ).The analog of Proposition3
is the following theorem, the proof of which is similar.Proposition 7.
For any fuzzy
setλ in an fts X, one has
(1)
sBdsBdλ≤sBdλ,(2)
sBdsBdsBdλ≤sBdsBdλ.Remark 7.
As in the case of Proposition3(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.Example 14.
Chooseλ={a.6,b.4} in fts X of Example 8. Then,(46)sBdλ={a.5,b.8}≰{a.5,b.2}=sBdsBdλ.Lemma 3 (see [13]).
Ifλ is a fuzzy set of X and μ is a fuzzy set of Y, then(47)1−(λ×μ)=λc×1∨1×μc.Using Lemma3, we have the following one.Lemma 4.
Letλ be a fuzzy semiclosed (resp., fuzzy semiopen
[13]) set of an fts X and μ a fuzzy semiclosed (resp., fuzzy semiopen) set
of an fts Y. Then, λ×μ is a fuzzy closed (resp., fuzzy semiopen) set
of the fuzzy product space X×Y.Using Lemma4, we have the following theorem.Theorem 11.
Ifλ is a fuzzy set of fts X and μ of fts Y, then
(1)
sClλ×sClμ≥sCl(λ×μ),(2)
sIntλ×sIntμ≤sInt(λ×μ).Moreover, we have the following one.Theorem 12.
LetX and Y be product-related fts's. Then, for a fuzzy
set λ of X and a fuzzy set μ of Y,
one has
(1)
sCl(λ×μ)=sClλ×sClμ,(2)
sInt(λ×μ)=sIntλ×sIntμ.Proof.
(1) For fuzzy setsλi's of X and μj's of Y, we first note that
(i)
inf{λi,μj}=min⁡{inf⁡λi,inf⁡μj},(ii)
inf{λi×1}=(inf⁡λi)×1,(iii)
inf{1×μj}=1×(inf⁡μj).
In view of
Theorem 11, it is sufficient to show that sCl(λ×μ)≥sClλ×sClμ. Let λi∈FSO(X) and μj∈FSO(Y). Then,(48)sCl(λ×μ)=inf⁡{(λi×μj)c∣(λi×μj)c≥λ×μ}=inf⁡{λic×1∨1×μjc∣λic×1∨1×μjc≥λ×μ}=inf⁡{λic×1∨1×μjc∣λic≥λorμjc≥μ}=min⁡(inf⁡{λic×1∨1×μjc∣λic≥λ},inf⁡{λic×1∨1×μjc∣μjc≥μ}).Since(49)inf⁡{λic×1∨1×μjc∣λic≥λ}≥inf⁡{λic×1∣λic≥λ}=inf⁡{λic∣λic≥λ}×1=(sClλ)×1,inf⁡{λic×1∨1×μjc∣μjc≥μ}≥inf⁡{μjc×1∣μjc≥μ}=1×inf⁡{μjc∣μjc≥μ}=1×(sClμ),we have(50)sCl(λ×μ)≥min⁡(sClλ×1,1×sClμ)=sClλ×sClμ.
(2) This follows from (1) using the facts that(sIntψ)c=sClψc and (sClψ)c=sIntψc.The analog of Theorem5
is the following, the proof of which is similar.Theorem 13.
LetXi,i=1,2,…,n, be a family of product-related fuzzy
topological spaces. If each λi is a fuzzy set in Xi, then(51)sBd∏i=1nλi=[sBdλ1×sClλ2×⋯×sClλn]∨[sClλ1×sBdλ2×sClλ3×⋯×sClλn]∨⋯∨[sClλ1×sClλ2×⋯×sBdλn].The following theorem gives a necessary condition for
fuzzy semicontinuous functions in terms of fuzzy boundary and fuzzy
semiboundary.Theorem 14.
Letf:X→Y be a fuzzy semicontinuous function. Then,
one has(52)sBdf−1(μ)≤f−1(Bdμ),for any fuzzy set μ in Y.Proof.
Letf be fuzzy semicontinuous and μ a fuzzy set in Y.
Then, Clμ is fuzzy closed in Y, which implies that f−1(Clμ) is fuzzy semiclosed in X. Therefore,
(53)sBdf−1(μ)=sClf−1(μ)∧sCl(f−1(μ))c≤sClf−1(Clμ)∧sClf−1(Clμc)=f−1(Clμ)∧f−1(Clμc)=f−1(Clμ∧Clμc)=f−1(Bdμ).Hence, sBdf−1(μ)≤f−1(Bdμ).Definition 17 (see [28]).
A functionf: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 Theorem14.Theorem 15.
Letf:X→Y be a fuzzy irresolute function. Then,
one has(54)sBdf−1(μ)≤f−1(sBdμ),for any fuzzy set μ in Y.Definition 18.
A fuzzy setλ 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 pointe=xα∈sClλ if each semi-Q-neighborhood of e is quasicoincident with λ.Proof.
xα∈sClλ if and only if for every fuzzy closed set ψ≥λ,xα∈ψ. This gives ψ(xα)≥λ(xα). Equivalently, xα∈sClλ if and only if for every fuzzy semiopen set β≤λc,β(xα)≤λc(xα). That is, for every fuzzy open set β satisfying β(xα)≥λc,β is not contained in λc,
or βqλcc=λ. Thus, xα∈sClλ if every fuzzy open Q-neighborhood β of xα is quasicoincident with λ.Definition 19.
A fuzzy pointe is called semiadherent point of a fuzzy set λ if every semi-Q-neighborhood of e is quasicoincident with λ.Definition 20.
A
fuzzy pointe 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 λsd. It is evident that λsd≤sClλ.Proposition 8.
Letλ be a fuzzy set in X,
then sClλ=λ∨λsd.Proof.
LetΩ={e∣eisasemiadherentpointofλ}. Then, from Theorem 16, sClλ=∨Ω. On the other hand, e∈Ω is either e∈λ or e∉λ;
for the latter case, by Definition 20, e∈λsd, hence sClλ=∨Ω≤λ∨λsd. The reverse inclusion is obvious.Corollary 3.
For any fuzzy setλ in an fts X,λ is fuzzy semiclosed if and only if λsd≤λ.Definition 21 (see [13]).
Letf: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 Corollary3 and characterize fuzzy
semicontinuous functions in terms of fuzzy semiderived set as
follows.Theorem 17.
Letf:X→Y be a function. Then, the following
conditions are equivalent:
(1)
f is fuzzy semicontinuous,(2)
f(λsd)≤Clf(λ), for any fuzzy set λ in X.Proof.
(1)⇒(2) Let f be fuzzy semicontinuous and λ a fuzzy set in X.
Since Clf(λ) is fuzzy closed in Y,f−1(Clf(λ)) is fuzzy semiclosed in X such that λ≤f−1(Clf(λ)). λ≤f−1(sClf(λ)) gives sClλ≤sClf−1(Clf(λ))=f−1(Clf(λ)). Therefore, f(λsd)≤f(sClλ)≤ff−1(Clf(λ))≤Clf(λ). Consequently, f(λsd)≤Clf(λ).
(2)⇒(1) Suppose f(λsd)≤Clf(λ).
Let μ be any fuzzy closed set in Y. We show that f−1(μ) is fuzzy semiclosed in X. By our hypothesis, f([f−1(μ)]sd)≤Clf(f−1(μ))≤Clμ=μ, f([f−1(μ)]sd)≤μ gives [f−1(μ)]sd≤f−1(f[f−1(μ)]sd)≤f−1(μ),
or [f−1(μ)]sd≤f−1(μ) implies that f−1(μ) is fuzzy semiclosed in X. Thus, f is fuzzy semicontinuous.Finally, we characterize fuzzy irresolute functions
via fuzzy semiderived set as follows.Theorem 18.
Letf:X→Y be a function. Then, the following
conditions  are equivalent:
(1)
f is fuzzy irresolute,(2)
f(λsd)≤sClf(λ), for any fuzzy set λ in X.



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*Source: 586893-2008-06-04.xml*