Advances in Fuzzy Systems

Volume 2008, Article ID 586893, 9 pages

http://dx.doi.org/10.1155/2008/586893

## Fuzzy Boundary and Fuzzy Semiboundary

^{1}Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, Pakistan^{2}Department of Mathematics, King Abdul Aziz University, P.O. Box 80203, Jeddah 21589, Kingdom of Saudi Arabia

Received 29 September 2007; Revised 3 February 2008; Accepted 21 April 2008

Academic Editor: Eyke Hüllermeier

Copyright © 2008 M. Athar and B. Ahmad. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 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 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 -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 set in a set is a function from to , that is, .

*Definition 1 (see [1]). * Let and be fuzzy sets in .
Then, for all ,

More generally, for a family of fuzzy sets in the union and intersection are defined by

The empty fuzzy set is defined as for all and the symbol denotes the fuzzy set , for all .

*Definition 2 (see [2]). *Let be a function. Let be a fuzzy set in with membership function .
Then, the inverse of ,
written as ,
is a fuzzy set in whose membership function is defined
byConversely, let be a fuzzy set in with membership function .
The image of ,
written as ,
is a fuzzy set in whose membership function is given
byfor all ,
where .

*Definition 3 (see [2]). *A fuzzy topology is a family of fuzzy sets in ,
which satisfies the following conditions:
(1)(2)if ,
then ;(3)if for each ,
then .

is called a fuzzy topology for , and the pair is a fuzzy topological space. Every member of is called -open fuzzy set (or simply fuzzy open set). A fuzzy set is -closed if and only if its complement is -open.

As in general topology, the indiscrete fuzzy topology contains only and , while the discrete fuzzy topology contains all fuzzy sets. In the sequel, we write an fts (or in place of “a space with fuzzy topology ”.

*Definition 4 (see [2]). *The closure and interior of
a fuzzy set in an fts 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 and in an fts one has the
following:
*(1)* is fuzzy closed (resp., fuzzy open) (resp., );*(2)* ();*(3)* ();*(4)(5)(6)(7)(8)(9)

*Definition 5 (see [2]). *A function is said to be fuzzy continuous if and only if
the inverse of each -fuzzy open set is -fuzzy open.

Theorem 1 (see [19, 26]). *Let be a function from an fts to another fts . Then, the following conditions are
equivalent:
*(1)* is fuzzy continuous,*(2)*the inverse of each -closed fuzzy set is -closed,*(3)*for each fuzzy set in , , *(4)*for each fuzzy set in , .*

#### 3. Fuzzy Boundary

*Definition 6 (see [24]). *Let be a fuzzy set in an fts . Then, the fuzzy boundary of is defined as Obviously, is a fuzzy closed set.

*Remark 1. *In classical topology, for an arbitrary set of a topological space ,
we have ,
but in fuzzy topology we have ,
for an arbitrary fuzzy set in 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 and in an fts , the following
conditions hold.*

(1)(2)*If is fuzzy closed, then .*(3)*If is fuzzy open, then .*(4)*Let and (resp., ) Then, (resp., ), where (resp., ) denotes the class of fuzzy closed (resp.,
fuzzy open) sets in .*(5)

*Proof. *(1)

(2) hence

(3) is fuzzy open implies is fuzzy closed. By (2), and by (1), we get

(4) Since implies we have since

(5)

The converse of (2) and (3) of Proposition 1 is, in general, not true as is shown by the following example.

*Example 1. *Let be a set and ,
the fuzzy topology generated by fuzzy sets and Then, Choose and Calculations give

The following proposition gives some more properties of fuzzy boundary.

Proposition 2. *Let be a fuzzy set in an fts .
Then,
*(1)(2)(3)(4)

*Proof. *(1) Since therefore we have Thus This proves (1),

(2)

(3)

(4)

The following example shows that the equality does not hold in Proposition 2(2)–(4).

*Example 2. *Choose and in Example 1. Then, calculations give

*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.

*Example 3. *In the fts of Example 1, we choose fuzzy set then calculations giveIt is easily seen that

Theorem 2. *Let and be fuzzy sets in an fts Then, *

*Proof. *

In Theorem 2, the equality does not hold as is shown by the following:

*Example 4. *In Example 1, choose fuzzy sets and Then, calculations giveThe following examples show
thatFor this, choose ,
and Then, calculations giveHowever, we have the following
theorem.

Theorem 3. *For any fuzzy sets and in an fts one
has*

*Proof. *

Corollary 1. *For any fuzzy sets and in an fts one
has*

*Example 5. *To show
that the equality in Theorem 3, in general, does not hold, choose and in the fts defined in Example 1. Then, calculations
give

In general topology, it is known thatfor any subset of a space However, in fuzzy topology, we have the following proposition.

Proposition 3. *For any fuzzy
set in an fts one has
*(1)(2)

*Proof. * (1) .

*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. *Choose in fts of Example 1. Then, but

*Definition 7 (see [27]). *If is a fuzzy set of and is a fuzzy set of then

*Definition 8 (see [13]). *An fts is product related to another fts if for any fuzzy set of and of whenever and imply where and there exist and such that and and

Theorem 4 (see [13]). *Let and be product-related fts's. Then, for a fuzzy
set of and a fuzzy set of one has
*(1)(2)

Lemma 2. *For fuzzy
sets , , ,
and in a set one
has*

*Proof. *

Theorem 5. *Let be a family of product-related fuzzy
topological spaces. If each is a fuzzy set in then*

*Proof. *We use Proposition 2(1),
Theorem 4, and Lemma 2 to prove this.

It suffices to prove this for Consider

Theorem 6. *Let be a fuzzy continuous function.
Then,**for any fuzzy set in .*

*Proof. *Let be fuzzy continuous and a fuzzy set in .
Then,

Therefore,

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 in is called a fuzzy point if and only if it
takes the value 0 for all except one, say, If its value at is one denotes this
fuzzy point by . The point is called support of fuzzy point and is denoted as

*Definition 10 (see [24]). *A fuzzy point is said to be quasicoincident (also written as
-coincident or -coincident) with a fuzzy set denoted by if for some ,
or

*Definition 11 (see [24]). *A fuzzy set in an fts is called a -neighborhood of a fuzzy point if there exists a such that

*Definition 12 (see [24]). *A fuzzy point is called an adherent point of a fuzzy set if every -neighborhood of is quasicoincident with

*Definition 13 (see [24]). *A fuzzy point is called an accumulation point of a fuzzy set if is an adherent point of and every -neighborhood of and is quasicoincident at some point different
from whenever The union of all the accumulation points of is called the derived set of denoted as It is evident that

Proposition 4 (see [24]). *Let be a fuzzy set in an fts ,
then
*(1)*;*(2)* is fuzzy closed if and only if *

We use Proposition 4 and prove the following theorem.

Theorem 7. *Let be a function. Then, the following
conditions are equivalent:
*(1)* is fuzzy continuous;*(2)* for any fuzzy set in .*

*Proof. *(1)(2) Let be fuzzy continuous and a fuzzy set in .
Since is fuzzy closed in then is fuzzy closed in . gives Therefore, Consequently,

(2)(1) Suppose .
Let be any fuzzy closed set in . We show that is fuzzy closed in By our hypothesis, gives or implies that is fuzzy closed in Thus, is fuzzy continuous.

#### 4. Fuzzy Semiboundary

First, we recall some notions.

*Definition 14 (see [13]). *Let be a fuzzy set in an fts .
Then, is called a fuzzy semiopen set of if there exists a such that .

*Definition 15 (see [22]). *Let be a fuzzy set in an fts .
Then, semiclosure (briefly ) and semi-interior (briefly ) of 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.

Theorem 8. *For fuzzy sets and in an fts ,
one has
*(1)*;*(2)*;*(3)*;*(4)*;*(5)*;*(6)*;*(7)*;*(8)*.*

*Proof. *(1) and are both fuzzy semiopen and , imply and .
Combining, or

(2) and imply and therefore Conversely and implies and is fuzzy semiopen. But is the largest fuzzy semiopen set contained in hence This gives the equality.

(3) It follows easily from (2).

(4) Since

(5)–(8) Proofs are straightforward.

The inequalities (1) and (4) of Theorem 8 are irreversible as is shown by the following example.

*Example 7. *Let be a set and the lattice of membership grades for fuzzy
sets in .
Let be the fuzzy topology on given asWe choose fuzzy sets ,
and Then, calculations give

*Definition 16 (see [25]). *Let be a fuzzy set in an fts Then, the fuzzy semiboundary of is defined as Obviously, is a fuzzy semiclosed set.

*Remark 5. *In fuzzy topology, we have ,
for an arbitrary fuzzy set in ,
the equality does not hold as the following example shows.

*Example 8. *Let be a set with fuzzy topology Choose then calculations give

In the following theorem, (1)–(5) are analogs of Proposition 1 and hence we omit their proofs.

Proposition 5. *For a fuzzy set in an fts the following
conditions hold.
*(1)(2)*If is fuzzy semiclosed, then *(3)*If is fuzzy semiopen, then *(4)*Let and (resp., ) Then, (resp., ), where (resp., ) denotes the class of fuzzy semiclosed
(resp., fuzzy semiopen) sets in *(5)(6)(7)

*Proof. *(6) Since and ,
then we have

(7)

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.

*Example 9. *Choose , , in the fts defined in Example 8. Then, calculations
give

The following is analog of Proposition 2 and hence we omit its proof.

Proposition 6. *Let be a fuzzy set in an fts .
Then, one has
*(1)(2)(3)(4)

To show that the inequalities (2), (3), and (4) of Proposition 6, are, in general, irreversible, we have the following example.

*Example 10. *Choose and in the fts defined in Example 8. Then, calculations
give

*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 of Example 8, we choose fuzzy set then calculations giveIt is easily seen that

Theorem 9. *Let and be fuzzy sets in an fts Then, *

The reverse inequality in Theorem 9 is, in general, not true as is shown by the following example.

*Example 12. *In Example 8, choose fuzzy sets and Then, calculations giveThe following example shows
thatFor this choose and Then, calculations giveHowever, we have the following
theorem which is an analog of Theorem 3.

Theorem 10. *For any fuzzy sets and in an fts one
has*

Corollary 2. *For any fuzzy sets and in an fts one has*

*Example 13. *To show that the reverse inequality in Theorem 10 is, in general, not true, choose
fuzzy sets and 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 in an fts one has
*(1)(2)

*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.

*Example 14. *Choose in fts of Example 8. Then,

Lemma 3 (see [13]). *If is a fuzzy set of and is a fuzzy set of then*

Using Lemma 3, we have the following one.

Lemma 4. *Let be a fuzzy semiclosed (resp., fuzzy semiopen
[13]) set of an fts and a fuzzy semiclosed (resp., fuzzy semiopen) set
of an fts Then, is a fuzzy closed (resp., fuzzy semiopen) set
of the fuzzy product space *

Using Lemma 4, we have the following theorem.

Theorem 11. *If is a fuzzy set of fts and of fts then
*(1)*, *(2)

Moreover, we have the following one.

Theorem 12. *Let and be product-related fts's. Then, for a fuzzy
set of and a fuzzy set of ,
one has
*(1)(2)

*Proof. *(1) For fuzzy sets of and of we first note that
(i)inf (ii)inf (iii)inf
In view of
Theorem 11, it is sufficient to show that Let and Then,Sincewe have

(2) This follows from (1) using the facts that and

The analog of Theorem 5 is the following, the proof of which is similar.

Theorem 13. *Let be a family of product-related fuzzy
topological spaces. If each is a fuzzy set in then*

The following theorem gives a necessary condition for fuzzy semicontinuous functions in terms of fuzzy boundary and fuzzy semiboundary.

Theorem 14. *Let be a fuzzy semicontinuous function. Then,
one has**for any fuzzy set in .*

*Proof. *Let be fuzzy semicontinuous and a fuzzy set in .
Then, is fuzzy closed in which implies that is fuzzy semiclosed in Therefore,

Hence,

*Definition 17 (see [28]). *A function is said to be fuzzy irresolute if is fuzzy semiopen in ,
for each fuzzy semiopen set in .

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 be a fuzzy irresolute function. Then,
one has**for any fuzzy set in .*

*Definition 18. *A fuzzy set in an fts is called a fuzzy semi--neighborhood of a
fuzzy point if there exists a fuzzy semiopen set in such that

Theorem 16. *A fuzzy point if each semi--neighborhood of is quasicoincident with *

*Proof. * if and only if for every fuzzy closed set This gives Equivalently, if and only if for every fuzzy semiopen set That is, for every fuzzy open set satisfying is not contained in ,
or Thus, if every fuzzy open -neighborhood of is quasicoincident with .

*Definition 19. *A fuzzy point is called semiadherent point of a fuzzy set if every semi--neighborhood of is quasicoincident with

*Definition 20. *A
fuzzy point is called a semiaccumulation point of a fuzzy
set if is a semiadherent point of and every semi--neighborhood of and is quasicoincident at some point different
from supp whenever The union of all the semiaccumulation points
of is called the fuzzy semiderived set of denoted as It is evident that

Proposition 8. *Let be a fuzzy set in ,
then .*

*Proof. *Let Then, from Theorem 16, On the other hand, is either or ;
for the latter case, by Definition 20, hence The reverse inclusion is obvious.

Corollary 3. *For any fuzzy set in an fts is fuzzy semiclosed if and only if *

*Definition 21 (see [13]). *Let be a function from an fts to another fts Then, is said to be fuzzy semicontinuous function if is fuzzy semiopen in for each fuzzy open set in

We use Corollary 3 and characterize fuzzy semicontinuous functions in terms of fuzzy semiderived set as follows.

Theorem 17. *Let be a function. Then, the following
conditions are equivalent:
*(1)* is fuzzy semicontinuous,*(2)* for any fuzzy set in .*

*Proof. * Let be fuzzy semicontinuous and a fuzzy set in .
Since is fuzzy closed in is fuzzy semiclosed in such that . gives Therefore, Consequently,

(2)(1) Suppose .
Let be any fuzzy closed set in Y. We show that is fuzzy semiclosed in By our hypothesis, , gives ,
or implies that is fuzzy semiclosed in Thus, is fuzzy semicontinuous.

Finally, we characterize fuzzy irresolute functions via fuzzy semiderived set as follows.

Theorem 18. *Let be a function. Then, the following
conditions are equivalent:
*(1)* is fuzzy irresolute,*(2)* for any fuzzy set in .*

#### Acknowledgment

The authors are grateful to the referees whose valuable suggestions have improved the paper a lot.

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