- Email: [email protected]
On fuzzy Λb sets and fuzzy Λb continuity
Chaos, Solitons and Fractals 42 (2009) 1024–1030
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
On fuzzy Kb sets and fuzzy Kb continuity Gülhan Aslım *, Gizem Günel Department of Mathematics, Faculty of Science, Ege University, Izmir, Turkey
a r t i c l e
i n f o
Article history: Accepted 26 February 2009
a b s t r a c t We introduce a new class of fuzzy open sets called fuzzy Kb sets which includes the class of fuzzy c-open sets due to Hanafy [Hanafy IM. Fuzzy c-open sets and fuzzy c-continuity. J Fuzzy Math 1999;7:419–30]. We also define a weaker form of fuzzy Kb sets termed as fuzzy locally Kb sets. By means of these new sets, we present the notions of fuzzy Kb continuity and fuzzy locally Kb continuity which are weaker than fuzzy c-continuity due to Hanafy [Hanafy IM. Fuzzy c-open sets and fuzzy c-continuity. J Fuzzy Math 1999;7:419– 30] and furthermore we investigate the relationships between these new types of continuity and some others. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction The concepts of fuzzy sets and fuzzy topology were firstly given by Zadeh in [13] and Chang in [4], and after then there have been many developments on defining uncertain situations and relations in more realistic way. The fuzzy topology theory has rapidly began to play an important role in many different scientific areas such as economics, quantum physics and geographic information system (GIS). For instance, Wenzheng Shi and Kimfung Liu mentioned that the fuzzy topology theory can potentially provide a more realistic description of uncertain spatial objects and uncertain relations in [11] where they developed the computational fuzzy topology which is based on the interior and the closure operator. Besides, the concepts of fuzzy topology and fuzzy sets have very important applications on particle physic in connection with string theory and 1 theory which were studied by El-Naschie [8–10]. In the fuzzy topology, the weaker forms of fuzzy open sets, which were constructed by the compositions of different combinations of the closure and interior operator, have been studied by several mathematicians [2,5–7,12]. In general topology, the class of b-open sets was presented in [1] and the class of Kb sets which includes the class of bopen sets was firstly studied in [3]. Then, Hanafy defined the class of fuzzy c-open sets as an extension of b-open sets to fuzzy topology in [5]. In this paper, we introduce fuzzy Kb sets including the class of fuzzy c-open sets due to Hanafy and establish several properties of fuzzy Kb sets. By means of these sets, we also define the class of fuzzy locally Kb sets, the notions of fuzzy Kb continuity and fuzzy locally Kb continuity. We discuss the relationships between these new types of continuity and other weaker types of fuzzy continuity. We think that the present paper may have applications not only on the computational fuzzy topology based in [11] but also on the particle physic in connection with string theory and 1 theory studied by El-Naschie [8–10]. 2. Preliminaries Throughout this paper, by ðX; sÞ we mean a fuzzy topological space in Chang’s [4] sense.
* Corresponding author. E-mail addresses: [email protected] (G. Aslım), [email protected] (G. Günel). 0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2009.02.042
G. Aslım, G. Günel / Chaos, Solitons and Fractals 42 (2009) 1024–1030
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Definition 2.1. Let A be a fuzzy set of a fuzzy topological space ðX; sÞ. A is called (a) a fuzzy semiopen [2] if A 6 clðintAÞ, (b) a fuzzy preopen set [6] if A 6 intðclAÞ, (c) a fuzzy semipreopen set [12] if A 6 clðintðclAÞÞ, (d) a fuzzy a-open set [7] if A 6 intðclðintAÞÞ. The complements of all these sets are called fuzzy semiclosed, preclosed, semipreclosed and a-closed sets, respectively. The family of all fuzzy semiopen sets (resp. fuzzy preopen, fuzzy semipreopen and fuzzy a-open sets) is denoted by FSOðXÞ (resp. FPOðXÞ; FSPOðXÞ and F aðXÞ). Definition 2.2 [5]. Let ðX; sÞ be a fuzzy topological space and A be a fuzzy set of X. Then (a) A is called a fuzzy c-open set if A 6 intðclAÞ _ clðintAÞ. (b) A is called a fuzzy c-closed set if intðclAÞ ^ clðintAÞ 6 A. The family of all fuzzy c-open and fuzzy c-closed sets of ðX; sÞ will be denoted by cOðXÞ and cCðXÞ, respectively. Definition 2.3 [5]. Let ðX; sÞ be a fuzzy topological space and A be a fuzzy set of X. The fuzzy c-interior and fuzzy c-closure of A are defined as follows, respectively:
_ intc ðAÞ ¼ fO : O 6 A; O 2 cOðXÞg; ^ clc ðAÞ ¼ fF : A 6 F; F 2 cCðXÞg: Remark 2.4.
fuzzy preopen %
&
&
%
fuzzy open ! fuzzy a-open
fuzzy c-open ! fuzzy semipreopen fuzzy semiopen
None of the implications can be reversed as shown in [5]. Lemma 2.5 [5]. Let ðX; sÞ be a fuzzy topological space. (a) Any union of fuzzy c-open sets is a fuzzy c-open set. (b) Any intersection of fuzzy c-closed sets is a fuzzy c-closed set.
Remark 2.6 [5]. For a fuzzy topological space the intersection of two fuzzy c-open sets may not be a fuzzy c-open set. 3. Fuzzy Kb and fuzzy V b sets In this section, we define the class of fuzzy Kb sets which is a weaker form of fuzzy c-open sets and investigate some basic properties of this class. We also present the fuzzy V b sets as the dual concept of fuzzy Kb sets. Definition 3.1. Let ðX; sÞ be a fuzzy topological space and B be a fuzzy set of X. The fuzzy BKb and BV b set of B are defined as follows:
^ fO : B 6 O; O 2 cOðXÞg; _ ¼ fF : F 6 B; F 2 cCðXÞg:
BKb ¼ BV b
Proposition 3.2. Let ðX; sÞ be a fuzzy topological space, A; B and Bk ðk 2 XÞ be the fuzzy sets of X. The following statements are valid: (a) B 6 BKb , (b) If A 6 B, then AKb 6 BKb , (c) ðBKb ÞKb ¼ BKb ,
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(d) (e) (f) (g) (h) (i) (j) (k)
G. Aslım, G. Günel / Chaos, Solitons and Fractals 42 (2009) 1024–1030
W K W K B b 6 B b, Vk2X k Kb k2 VX k Kb 6 k2X Bk , k2X Bk BV b 6 B, If A 6 B, then AV b 6 BV b , ðBV b ÞV b ¼ BV b , W V W Vb B b, k2 Vk2X Bk 6 VX k V b Vb 6 k2X Bk , k2X Bk ðBc ÞKb ¼ ðBV b Þc .
Proof. We will prove only (e) and (k). The others can be proved in a similar way. For all k 2 X,
^
Bk 6 Bk )
k2X
^
!Kb 6 ðBk ÞKb
Bk
k2X
)
^
!Kb Bk
6
k2X
^
K
Bk b
k2X
which proves (e). For (k),
ðBV b Þc ¼
_
c fF : F 6 B; F 2 cCðXÞg
^ c c fF : B 6 F c ; F c 2 cOðXÞg ^ ¼ fO : Bc 6 O; O 2 cOðXÞg ¼ ðBc ÞKb : ¼
Definition 3.3. Let B be a fuzzy set of a fuzzy topological space ðX; sÞ. Then B is called (a) a fuzzy Kb set if B ¼ BKb , (b) a fuzzy V b set if B ¼ BV b . The family of all fuzzy Kb sets and V b sets will be denoted by Kb ðXÞ and V b ðXÞ, respectively. Theorem 3.4. B is a fuzzy Kb set if and only if Bc is a fuzzy V b set. Proof. It is obvious.
h
Proposition 3.5. Let A be fuzzy set of a fuzzy topological space ðX; sÞ. (a) If A 2 cOðXÞ, then A 2 Kb ðXÞ. (b) If A 2 cCðXÞ, then A 2 V b ðXÞ.
Proof. It is obvious.
h
Remark 3.6. None of the reverse of implications in Proposition 3.5 is valid as shown in the following example. Example 3.7. (a) Let X ¼ fa; bg and A; B be the fuzzy sets of X defined as follows:
AðaÞ ¼ 0:2;
BðaÞ ¼ 0:1;
AðbÞ ¼ 0:3;
BðbÞ ¼ 0:4:
Consider the fuzzy topology
s ¼ f0; 1; Ag. Clearly, it can be shown that clB ¼ Ac ; intðclBÞ ¼ A; intB ¼ 0; clðintBÞ ¼ 0. Since
BiintðclBÞ _ clðintBÞ ¼ maxf0; Ag ¼ A; B is not a fuzzy c-open set. However, BKb ¼ B. Therefore, B is a fuzzy Kb set. (b) Let X ¼ fa; bg and A; B be the fuzzy sets of X defined as follows:
AðaÞ ¼ 0:2;
BðaÞ ¼ 0:3;
AðbÞ ¼ 0:8;
BðbÞ ¼ 0:9:
G. Aslım, G. Günel / Chaos, Solitons and Fractals 42 (2009) 1024–1030
Consider the fuzzy topology It is clear that
clB ¼ 1;
s ¼ f0; 1; Ag.
intðclBÞ ¼ 1;
intB ¼ A;
clðintBÞ ¼ 1
and
1 ¼ minf1; 1g ¼ intðclBÞ ^ clðintBÞiB: Hence B is not a fuzzy c-closed set, but BV b ¼ B, so B 2 V b ðXÞ.
Remark 3.8. The following diagram of the implications is true.
fuzzy preopen %
fuzzy semipreopen &
fuzzy open ! fuzzy a-open
% fuzzy c-open
&
% fuzzy semiopen
& fuzzy Kb set
Fuzzy Kb sets and fuzzy semipreopen sets are independent as shown in the following examples. Example 3.9. Let X ¼ fa; bg,
AðaÞ ¼ 0:2;
BðaÞ ¼ 0:7;
AðbÞ ¼ 0:6;
BðbÞ ¼ 0:3
and the fuzzy topology on X be
clB ¼ Ac ;
intðclBÞ ¼ 0;
s ¼ f0; 1; Ag. It can easily shown that clðintðclBÞÞ ¼ 0:
Since BiclðintðclBÞÞ, B is not a fuzzy semipreopen set. On the other hand BKb ¼ B. That is B is a fuzzy Kb set. Example 3.10. Consider the set X ¼ fa; bg, the fuzzy sets
AðaÞ ¼ 0:2; AðbÞ ¼ 0;
BðaÞ ¼ 0:1; BðbÞ ¼ 1
and let the fuzzy topology on X be c
clB ¼ A ;
s ¼ f0; 1; Ag. Since
clðintðclBÞÞ ¼ Ac
intðclBÞ ¼ A;
and B 6 clðintðclBÞÞ ¼ Ac ; B is a fuzzy semipreopen set. But BKb –B, for BKb ðaÞ ¼ 0:2; BKb ðbÞ ¼ 1. Hence B R Kb ðXÞ. Theorem 3.11. Let A and Ak ðk 2 XÞ be the fuzzy sets of the fuzzy topological space ðX; sÞ. Then, (a) (b) (c) (d)
AKb is a fuzzy Kb set. AV b is a fuzzy V b set. V If fAk : k 2 Xg # Kb ðXÞ, then k2X Ak is a fuzzy Kb set. W If fAk : k 2 Xg # V b ðXÞ, then k2X Ak is a fuzzy V b set.
Proof. (a) By Proposition 3.2(c), ðAKb ÞKb ¼ AKb . Hence AKb is a fuzzy Kb set. (b) It is clear from Proposition 3.2(h). (c) For all k 2 X,
ðAk ÞKb ¼ Ak )
^
K
Ak b ¼
k2X
)
^ k2X
^
Ak
k2X
!Kb Ak
6
^ k2X
ðAk ÞKb ¼
^ k2X
Ak :
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V Kb V Kb V V Since for all k 2 X; k2X Ak 6 always holds, k2X Ak ¼ . k2X Ak k2X Ak (d) It can be proved in similar manner in (c). 4. Fuzzy locally Kb sets In this section, we introduce the class of fuzzy locally Kb sets including the class of fuzzy Kb sets, and give two characterizations of these sets. Definition 4.1. Let A be a fuzzy set of a fuzzy topological space ðX; sÞ. A is called fuzzy locally Kb set if there exists a fuzzy Kb set U and a fuzzy c-closed set B such that A ¼ U ^ B. Remark 4.2. Since A ¼ A ^ 1X for every fuzzy set A, every fuzzy Kb set is a fuzzy locally Kb set and every fuzzy c-closed set is a fuzzy locally Kb set. The following examples show that the converses are not true. Example 4.3. Consider the fuzzy topology and the fuzzy set B in Example 3.10. We have seen that B is not a fuzzy Kb set, but it is a fuzzy locally Kb set, since it can be represented as B ¼ U 1 ^ U 2 where
U 1 ðaÞ ¼ 0:2; U 1 ðbÞ ¼ 1;
U 2 ðaÞ ¼ 0:1; U 2 ðbÞ ¼ 1:
It can be easily shown that U 1 is a fuzzy Kb set and U 2 is a fuzzy c-closed set. Example 4.4. Let X ¼ fa; bg and
AðaÞ ¼ 0:4;
BðaÞ ¼ 1;
AðbÞ ¼ 0:8;
BðbÞ ¼ 0:9:
Now consider the fuzzy topology s ¼ f0; 1; Ag. It can be easily shown that B is a fuzzy c-open, and hence it is a fuzzy locally Kb set. On the contrary, it is not a fuzzy c-closed set. Theorem 4.5. Let A be a fuzzy set of a fuzzy topological space ðX; sÞ. The followings are equivalent: (a) A is a fuzzy locally Kb set. (b) A ¼ U ^ clc ðAÞ for a fuzzy Kb set U. (c) A ¼ AKb ^ clc ðAÞ. Proof. (a) ) (b): Let A ¼ U ^ B where U is a fuzzy Kb set and B is a fuzzy c-closed set. Since A 6 U and A 6 clc ðAÞ, we have A 6 U ^ clc ðAÞ. On the other hand, A 6 B and A 6 clc ðAÞ 6 clc ðBÞ ¼ B; U ^ clc ðAÞ 6 A which completes the proof. (b) ) (c): If A ¼ U ^ clc ðAÞ for a fuzzy Kb set U, then A 6 U. Thus, AKb 6 U Kb ¼ U which implies AKb ^ clc ðAÞ 6 U ^ clc ðAÞ ¼ A. Since A 6 AKb and A 6 clc ðAÞ; A 6 AKb ^ clc ðAÞ. (c) ) (a): Since AKb is a fuzzy Kb set and clc ðAÞ is a fuzzy c-closed set, A is a fuzzy locally Kb set. h 5. Fuzzy Kb continuity and fuzzy locally Kb continuity In this section we present two weaker forms of fuzzy continuity named fuzzy Kb continuity and fuzzy locally Kb continuity via the fuzzy Kb sets and fuzzy locally Kb sets and we obtain some characterizations of these new continuities. Definition 5.1. Let f : ðX; s1 Þ ! ðY; s2 Þ be a function from a fuzzy topological space ðX; s1 Þ into a fuzzy topological space ðY; s2 Þ. The function f is called (a) fuzzy continuous [4] if f 1 ðBÞ is a fuzzy open set of X for each B 2 s2 , (b) fuzzy semicontinuous [2] if f 1 ðBÞ is a fuzzy semiopen set of X for each B 2 s2 , (c) fuzzy precontinuous [6] if f 1 ðBÞ is a fuzzy preopen set of X for each B 2 s2 , (d) fuzzy strongly semicontinuous [7] if f 1 ðBÞ is a fuzzy a-open set of X for each B 2 s2 , (e) fuzzy c-continuous [5] if f 1 ðBÞ is a fuzzy c-open set of X for each B 2 s2 , (f) fuzzy semiprecontinuous [12] if f 1 ðBÞ is a fuzzy semipreopen set of X for each B 2 s2 , (g) fuzzy Kb continuous if f 1 ðBÞ is a fuzzy Kb set of X for each B 2 s2 . (h) fuzzy locally Kb continuous if f 1 ðBÞ is a fuzzy locally Kb set of X for each B 2 s2 .
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Remark 5.2. It is clear from the Remarks 3.8 and 4.2 that the implications of the following diagram hold.
fuzzy
fuzzy
precontinous % fuzzy continuous
!
semiprecontinuous &
%
fuzzy strongly
fuzzy
semicontinuous
c-continuous &
%
&
fuzzy
fuzzy
semicontinuous
Kb continuous # fuzzy locally Kb continuous
However, none of the implications of this diagram is reversed as shown in the following examples. Example 5.3. Let X ¼ fa; bg and the fuzzy topology s on X defined as in Example 3.7(a), Y ¼ f1; 2g; Cð1Þ ¼ 0:1; Cð2Þ ¼ 0:4 and the fuzzy topology on Y be # ¼ f0; 1; Cg. Now consider the map f : ðX; sÞ ! ðY; #Þ, defined as follows: f ðaÞ ¼ 1; f ðbÞ ¼ 2. Since
f 1 ðCÞðaÞ ¼ Cðf ðaÞÞ ¼ Cð1Þ ¼ 0:1; f 1 ðCÞðbÞ ¼ Cðf ðbÞÞ ¼ Cð2Þ ¼ 0:4; as shown in Example 3.7(a), f 1 ðCÞ ¼ B R cOðXÞ, but f 1 ðCÞ ¼ B 2 Kb ðXÞ. f is fuzzy Kb continuous, but not fuzzy c-continuous and hence f is not fuzzy semicontinuous, precontinuous, fuzzy strongly semicontinuous and continuous. Example 5.4. Let X ¼ fa; bg and the fuzzy topology s on X defined as in Example 3.10, and Y ¼ f1; 2g and the fuzzy set C on Y defined as follows: Cð1Þ ¼ 0:1; Cð2Þ ¼ 1. Let the fuzzy topology on Y be # ¼ f0; 1; Cg and the map f : ðX; sÞ ! ðY; #Þ defined as follows: f ðaÞ ¼ 1, f ðbÞ ¼ 2. Since
f 1 ðCÞðaÞ ¼ Cðf ðaÞÞ ¼ Cð1Þ ¼ 0:1; f 1 ðCÞðbÞ ¼ Cðf ðbÞÞ ¼ Cð2Þ ¼ 1; as shown in Example 4.3 f 1 ðCÞ ¼ B is not a fuzzy Kb set but it is a fuzzy locally Kb set. Thus, f is not a fuzzy Kb continuous but it is a fuzzy locally Kb continuous function. Theorem 5.5. Let f : ðX; s1 Þ ! ðY; s2 Þ be a function from a fuzzy topological space ðX; s1 Þ into a fuzzy topological space ðY; s2 Þ. The following statements are equivalent: (a) f is fuzzy Kb continuous. (b) For all Bc 2 s2 ; f 1 ðBÞ 2 V b ðXÞ. (c) For all fuzzy set A of Y; ðf 1 ðintAÞÞKb 6 f 1 ðAÞ. Proof. (a) ) (b): Let Bc 2 s2 . Since f is Kb continuous, f 1 ðBc Þ ¼ ðf 1 ðBÞÞc 2 Kb ðXÞ. Thus, f 1 ðBÞ 2 V b ðXÞ. (b) ) (a): It can be proved in the above manner. (a) ) (c): Since intðAÞ 2 s2 and f is Kb continuous,
ðf 1 ðintAÞÞKb ¼ f 1 ðintAÞ 6 f 1 ðAÞ: (c) ) (a): Let B 2 s2 . Then intB ¼ B. By assumption,
ðf 1 ðBÞÞKb ¼ ðf 1 ðintBÞÞKb 6 f 1 ðBÞ: Since f 1 ðBÞ 6 ðf 1 ðBÞÞKb always holds, f 1 ðBÞ 2 Kb ðXÞ. h Theorem 5.6. Let f : ðX; s1 Þ ! ðY; s2 Þ be a function from a fuzzy topological space ðX; s1 Þ into a fuzzy topological space ðY; s2 Þ. The following statements are equivalent: (a) f is fuzzy locally Kb continuous. V (b) For all fuzzy set A of Y; f 1 ðintAÞ ¼ ðf 1 ðintAÞÞKb clc ðf 1 ðintAÞÞ. Vb W 1 1 int c ðf 1 ðclAÞÞ. (c) For all fuzzy set A of Y; f ðclAÞ ¼ ðf ðclAÞÞ
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Proof. (a) , (b): Since intA is a fuzzy open set, the proof is immediate from Theorem 4.5. (a) ) (c): ðclAÞc is a fuzzy open set, so by Theorem 4.5,
f 1 ððclAÞc Þ ¼ ðf 1 ðclAÞÞc ¼ ððf 1 ðclAÞÞc ÞKb
^
clc ððf 1 ðclAÞÞc Þ ^ ¼ ððf 1 ðclAÞÞ Þ ðintc ðf 1 ðclAÞÞÞc ; Vb c
W f 1 ðclAÞ ¼ ðf 1 ðclAÞÞV b int c ðf 1 ðclAÞÞ. (c) ) (a): Let A be a fuzzy open set. Thus, clðAc Þ ¼ Ac . By assumption,
_ f 1 ðclðAc ÞÞ ¼ f 1 ðAc Þ ¼ ðf 1 ðclðAc ÞÞÞV b intc ðf 1 ðclðAc ÞÞÞ _ c V ¼ ðf 1 ðA ÞÞ b intc ðf 1 ðAc ÞÞ _ ¼ ððf 1 ðAÞÞc ÞV b int c ððf 1 ðAÞÞc Þ _ ¼ ððf 1 ðAÞÞKb Þc ðclc ððf 1 ðAÞÞÞÞc ^ clc ððf 1 ðAÞÞÞÞc : ¼ ððf 1 ðAÞÞKb Hence, f 1 ðAÞ ¼ f 1 ðAÞÞKb
V
clc ððf 1 ðAÞÞÞ which means f 1 ðAÞ is a fuzzy locally Kb set. h
Theorem 5.7. Let f : X ! Y be a fuzzy Kb continuous function and g : Y ! Z fuzzy continuous function. Then g f : X ! Z is a fuzzy Kb continuous function. Proof. It is clear from the equality ðg f Þ1 ðBÞ ¼ f 1 ðg 1 ðBÞÞ.
h
Theorem 5.8. Let f : X ! Y be a fuzzy continuous function. If g : X ! X Y the graph map of f is fuzzy Kb continuous, then f is a fuzzy Kb continuous function. Proof. Let B an open set of Y. Then 1X B is an open set of X Y. By Lemma 2.4. in [2],
g 1 ð1X BÞ ¼ 1X ^ f 1 ðBÞ ¼ f 1 ðBÞ 2 Kb ðXÞ:
6. Conclusion The significance of fuzzy topology has rapidly been appeared in many fields of both pure and applied sciences. The notions of fuzzy continuity and fuzzy open sets are fundamental structures of fuzzy topology. That is why the weaker forms of fuzzy continuity and open sets have been studied by many mathematicians. The weaker types of fuzzy open sets and fuzzy continuity, established in this paper, may play an important role in computational fuzzy topology which was developed in [11] and may have applications in quantum physics, particularly in connection with string theory and 1 theory [8–10]. References Andrijevic´ D. On b-open sets. Mat Vesnik 1996;48:59–64. Azad KK. On fuzzy semicontinuity, fuzzy almost continuity and fuzzy weakly continuity. J Math Anal Appl 1981;82:14–32. Caldas M, Jafari S, Noiri T. On Kb -sets and the associated topology sKb . Acta Math Hung 2006;110(4):337–45. Chang CL. Fuzzy topological spaces. J Math Anal Appl 1968;24:182–90. Hanafy IM. Fuzzy c-open sets and fuzzy c-continuity. J Fuzzy Math 1999;7:419–30. Singal MK, Prakash N. Fuzzy preopen sets and fuzzy preseparation axioms. Fuzzy Sets Syst 1991;44:273–81. Bin Shahna AS. On fuzzy strong semicontinuity and fuzzy precontinuity. Fuzzy Sets Syst 1991;44:303–8. El Naschie MS. On the uncertainty of cantorian geometry and the two-slit experiment. Chaos, Solitons and Fractals 1998;9(3):517–29. El Naschie MS. On the unification of heterotic strings, M theory and 1 theory. Chaos, Solitons and Fractals 2000;11(14):2397–408. El Naschie MS. On a fuzzy Kahler-like manifold which is consistent with the two slit experiment. Int J Nonlinear Sci Numer Simul 2005;6(2):94–7. Shi Wenzhong, Liu Kimfung. A fuzzy topology for computing the interior, boundary, and exterior of spatial objects quantitatively in GIS. Comput Geosci 2007;33:898–915. [12] Thakur SS, Singh S. On fuzzy semi-preopen sets and fuzzy semi-precontinuity. Fuzzy Sets Syst 1998;98:383–91. [13] Zadeh LA. Fuzzy sets. Inform Control 1965;8:338–53. [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
On fuzzy Kb sets and fuzzy Kb continuity Gülhan Aslım *, Gizem Günel Department of Mathematics, Faculty of Science, Ege University, Izmir, Turkey
a r t i c l e
i n f o
Article history: Accepted 26 February 2009
a b s t r a c t We introduce a new class of fuzzy open sets called fuzzy Kb sets which includes the class of fuzzy c-open sets due to Hanafy [Hanafy IM. Fuzzy c-open sets and fuzzy c-continuity. J Fuzzy Math 1999;7:419–30]. We also define a weaker form of fuzzy Kb sets termed as fuzzy locally Kb sets. By means of these new sets, we present the notions of fuzzy Kb continuity and fuzzy locally Kb continuity which are weaker than fuzzy c-continuity due to Hanafy [Hanafy IM. Fuzzy c-open sets and fuzzy c-continuity. J Fuzzy Math 1999;7:419– 30] and furthermore we investigate the relationships between these new types of continuity and some others. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction The concepts of fuzzy sets and fuzzy topology were firstly given by Zadeh in [13] and Chang in [4], and after then there have been many developments on defining uncertain situations and relations in more realistic way. The fuzzy topology theory has rapidly began to play an important role in many different scientific areas such as economics, quantum physics and geographic information system (GIS). For instance, Wenzheng Shi and Kimfung Liu mentioned that the fuzzy topology theory can potentially provide a more realistic description of uncertain spatial objects and uncertain relations in [11] where they developed the computational fuzzy topology which is based on the interior and the closure operator. Besides, the concepts of fuzzy topology and fuzzy sets have very important applications on particle physic in connection with string theory and 1 theory which were studied by El-Naschie [8–10]. In the fuzzy topology, the weaker forms of fuzzy open sets, which were constructed by the compositions of different combinations of the closure and interior operator, have been studied by several mathematicians [2,5–7,12]. In general topology, the class of b-open sets was presented in [1] and the class of Kb sets which includes the class of bopen sets was firstly studied in [3]. Then, Hanafy defined the class of fuzzy c-open sets as an extension of b-open sets to fuzzy topology in [5]. In this paper, we introduce fuzzy Kb sets including the class of fuzzy c-open sets due to Hanafy and establish several properties of fuzzy Kb sets. By means of these sets, we also define the class of fuzzy locally Kb sets, the notions of fuzzy Kb continuity and fuzzy locally Kb continuity. We discuss the relationships between these new types of continuity and other weaker types of fuzzy continuity. We think that the present paper may have applications not only on the computational fuzzy topology based in [11] but also on the particle physic in connection with string theory and 1 theory studied by El-Naschie [8–10]. 2. Preliminaries Throughout this paper, by ðX; sÞ we mean a fuzzy topological space in Chang’s [4] sense.
* Corresponding author. E-mail addresses: [email protected] (G. Aslım), [email protected] (G. Günel). 0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2009.02.042
G. Aslım, G. Günel / Chaos, Solitons and Fractals 42 (2009) 1024–1030
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Definition 2.1. Let A be a fuzzy set of a fuzzy topological space ðX; sÞ. A is called (a) a fuzzy semiopen [2] if A 6 clðintAÞ, (b) a fuzzy preopen set [6] if A 6 intðclAÞ, (c) a fuzzy semipreopen set [12] if A 6 clðintðclAÞÞ, (d) a fuzzy a-open set [7] if A 6 intðclðintAÞÞ. The complements of all these sets are called fuzzy semiclosed, preclosed, semipreclosed and a-closed sets, respectively. The family of all fuzzy semiopen sets (resp. fuzzy preopen, fuzzy semipreopen and fuzzy a-open sets) is denoted by FSOðXÞ (resp. FPOðXÞ; FSPOðXÞ and F aðXÞ). Definition 2.2 [5]. Let ðX; sÞ be a fuzzy topological space and A be a fuzzy set of X. Then (a) A is called a fuzzy c-open set if A 6 intðclAÞ _ clðintAÞ. (b) A is called a fuzzy c-closed set if intðclAÞ ^ clðintAÞ 6 A. The family of all fuzzy c-open and fuzzy c-closed sets of ðX; sÞ will be denoted by cOðXÞ and cCðXÞ, respectively. Definition 2.3 [5]. Let ðX; sÞ be a fuzzy topological space and A be a fuzzy set of X. The fuzzy c-interior and fuzzy c-closure of A are defined as follows, respectively:
_ intc ðAÞ ¼ fO : O 6 A; O 2 cOðXÞg; ^ clc ðAÞ ¼ fF : A 6 F; F 2 cCðXÞg: Remark 2.4.
fuzzy preopen %
&
&
%
fuzzy open ! fuzzy a-open
fuzzy c-open ! fuzzy semipreopen fuzzy semiopen
None of the implications can be reversed as shown in [5]. Lemma 2.5 [5]. Let ðX; sÞ be a fuzzy topological space. (a) Any union of fuzzy c-open sets is a fuzzy c-open set. (b) Any intersection of fuzzy c-closed sets is a fuzzy c-closed set.
Remark 2.6 [5]. For a fuzzy topological space the intersection of two fuzzy c-open sets may not be a fuzzy c-open set. 3. Fuzzy Kb and fuzzy V b sets In this section, we define the class of fuzzy Kb sets which is a weaker form of fuzzy c-open sets and investigate some basic properties of this class. We also present the fuzzy V b sets as the dual concept of fuzzy Kb sets. Definition 3.1. Let ðX; sÞ be a fuzzy topological space and B be a fuzzy set of X. The fuzzy BKb and BV b set of B are defined as follows:
^ fO : B 6 O; O 2 cOðXÞg; _ ¼ fF : F 6 B; F 2 cCðXÞg:
BKb ¼ BV b
Proposition 3.2. Let ðX; sÞ be a fuzzy topological space, A; B and Bk ðk 2 XÞ be the fuzzy sets of X. The following statements are valid: (a) B 6 BKb , (b) If A 6 B, then AKb 6 BKb , (c) ðBKb ÞKb ¼ BKb ,
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(d) (e) (f) (g) (h) (i) (j) (k)
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W K W K B b 6 B b, Vk2X k Kb k2 VX k Kb 6 k2X Bk , k2X Bk BV b 6 B, If A 6 B, then AV b 6 BV b , ðBV b ÞV b ¼ BV b , W V W Vb B b, k2 Vk2X Bk 6 VX k V b Vb 6 k2X Bk , k2X Bk ðBc ÞKb ¼ ðBV b Þc .
Proof. We will prove only (e) and (k). The others can be proved in a similar way. For all k 2 X,
^
Bk 6 Bk )
k2X
^
!Kb 6 ðBk ÞKb
Bk
k2X
)
^
!Kb Bk
6
k2X
^
K
Bk b
k2X
which proves (e). For (k),
ðBV b Þc ¼
_
c fF : F 6 B; F 2 cCðXÞg
^ c c fF : B 6 F c ; F c 2 cOðXÞg ^ ¼ fO : Bc 6 O; O 2 cOðXÞg ¼ ðBc ÞKb : ¼
Definition 3.3. Let B be a fuzzy set of a fuzzy topological space ðX; sÞ. Then B is called (a) a fuzzy Kb set if B ¼ BKb , (b) a fuzzy V b set if B ¼ BV b . The family of all fuzzy Kb sets and V b sets will be denoted by Kb ðXÞ and V b ðXÞ, respectively. Theorem 3.4. B is a fuzzy Kb set if and only if Bc is a fuzzy V b set. Proof. It is obvious.
h
Proposition 3.5. Let A be fuzzy set of a fuzzy topological space ðX; sÞ. (a) If A 2 cOðXÞ, then A 2 Kb ðXÞ. (b) If A 2 cCðXÞ, then A 2 V b ðXÞ.
Proof. It is obvious.
h
Remark 3.6. None of the reverse of implications in Proposition 3.5 is valid as shown in the following example. Example 3.7. (a) Let X ¼ fa; bg and A; B be the fuzzy sets of X defined as follows:
AðaÞ ¼ 0:2;
BðaÞ ¼ 0:1;
AðbÞ ¼ 0:3;
BðbÞ ¼ 0:4:
Consider the fuzzy topology
s ¼ f0; 1; Ag. Clearly, it can be shown that clB ¼ Ac ; intðclBÞ ¼ A; intB ¼ 0; clðintBÞ ¼ 0. Since
BiintðclBÞ _ clðintBÞ ¼ maxf0; Ag ¼ A; B is not a fuzzy c-open set. However, BKb ¼ B. Therefore, B is a fuzzy Kb set. (b) Let X ¼ fa; bg and A; B be the fuzzy sets of X defined as follows:
AðaÞ ¼ 0:2;
BðaÞ ¼ 0:3;
AðbÞ ¼ 0:8;
BðbÞ ¼ 0:9:
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Consider the fuzzy topology It is clear that
clB ¼ 1;
s ¼ f0; 1; Ag.
intðclBÞ ¼ 1;
intB ¼ A;
clðintBÞ ¼ 1
and
1 ¼ minf1; 1g ¼ intðclBÞ ^ clðintBÞiB: Hence B is not a fuzzy c-closed set, but BV b ¼ B, so B 2 V b ðXÞ.
Remark 3.8. The following diagram of the implications is true.
fuzzy preopen %
fuzzy semipreopen &
fuzzy open ! fuzzy a-open
% fuzzy c-open
&
% fuzzy semiopen
& fuzzy Kb set
Fuzzy Kb sets and fuzzy semipreopen sets are independent as shown in the following examples. Example 3.9. Let X ¼ fa; bg,
AðaÞ ¼ 0:2;
BðaÞ ¼ 0:7;
AðbÞ ¼ 0:6;
BðbÞ ¼ 0:3
and the fuzzy topology on X be
clB ¼ Ac ;
intðclBÞ ¼ 0;
s ¼ f0; 1; Ag. It can easily shown that clðintðclBÞÞ ¼ 0:
Since BiclðintðclBÞÞ, B is not a fuzzy semipreopen set. On the other hand BKb ¼ B. That is B is a fuzzy Kb set. Example 3.10. Consider the set X ¼ fa; bg, the fuzzy sets
AðaÞ ¼ 0:2; AðbÞ ¼ 0;
BðaÞ ¼ 0:1; BðbÞ ¼ 1
and let the fuzzy topology on X be c
clB ¼ A ;
s ¼ f0; 1; Ag. Since
clðintðclBÞÞ ¼ Ac
intðclBÞ ¼ A;
and B 6 clðintðclBÞÞ ¼ Ac ; B is a fuzzy semipreopen set. But BKb –B, for BKb ðaÞ ¼ 0:2; BKb ðbÞ ¼ 1. Hence B R Kb ðXÞ. Theorem 3.11. Let A and Ak ðk 2 XÞ be the fuzzy sets of the fuzzy topological space ðX; sÞ. Then, (a) (b) (c) (d)
AKb is a fuzzy Kb set. AV b is a fuzzy V b set. V If fAk : k 2 Xg # Kb ðXÞ, then k2X Ak is a fuzzy Kb set. W If fAk : k 2 Xg # V b ðXÞ, then k2X Ak is a fuzzy V b set.
Proof. (a) By Proposition 3.2(c), ðAKb ÞKb ¼ AKb . Hence AKb is a fuzzy Kb set. (b) It is clear from Proposition 3.2(h). (c) For all k 2 X,
ðAk ÞKb ¼ Ak )
^
K
Ak b ¼
k2X
)
^ k2X
^
Ak
k2X
!Kb Ak
6
^ k2X
ðAk ÞKb ¼
^ k2X
Ak :
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V Kb V Kb V V Since for all k 2 X; k2X Ak 6 always holds, k2X Ak ¼ . k2X Ak k2X Ak (d) It can be proved in similar manner in (c). 4. Fuzzy locally Kb sets In this section, we introduce the class of fuzzy locally Kb sets including the class of fuzzy Kb sets, and give two characterizations of these sets. Definition 4.1. Let A be a fuzzy set of a fuzzy topological space ðX; sÞ. A is called fuzzy locally Kb set if there exists a fuzzy Kb set U and a fuzzy c-closed set B such that A ¼ U ^ B. Remark 4.2. Since A ¼ A ^ 1X for every fuzzy set A, every fuzzy Kb set is a fuzzy locally Kb set and every fuzzy c-closed set is a fuzzy locally Kb set. The following examples show that the converses are not true. Example 4.3. Consider the fuzzy topology and the fuzzy set B in Example 3.10. We have seen that B is not a fuzzy Kb set, but it is a fuzzy locally Kb set, since it can be represented as B ¼ U 1 ^ U 2 where
U 1 ðaÞ ¼ 0:2; U 1 ðbÞ ¼ 1;
U 2 ðaÞ ¼ 0:1; U 2 ðbÞ ¼ 1:
It can be easily shown that U 1 is a fuzzy Kb set and U 2 is a fuzzy c-closed set. Example 4.4. Let X ¼ fa; bg and
AðaÞ ¼ 0:4;
BðaÞ ¼ 1;
AðbÞ ¼ 0:8;
BðbÞ ¼ 0:9:
Now consider the fuzzy topology s ¼ f0; 1; Ag. It can be easily shown that B is a fuzzy c-open, and hence it is a fuzzy locally Kb set. On the contrary, it is not a fuzzy c-closed set. Theorem 4.5. Let A be a fuzzy set of a fuzzy topological space ðX; sÞ. The followings are equivalent: (a) A is a fuzzy locally Kb set. (b) A ¼ U ^ clc ðAÞ for a fuzzy Kb set U. (c) A ¼ AKb ^ clc ðAÞ. Proof. (a) ) (b): Let A ¼ U ^ B where U is a fuzzy Kb set and B is a fuzzy c-closed set. Since A 6 U and A 6 clc ðAÞ, we have A 6 U ^ clc ðAÞ. On the other hand, A 6 B and A 6 clc ðAÞ 6 clc ðBÞ ¼ B; U ^ clc ðAÞ 6 A which completes the proof. (b) ) (c): If A ¼ U ^ clc ðAÞ for a fuzzy Kb set U, then A 6 U. Thus, AKb 6 U Kb ¼ U which implies AKb ^ clc ðAÞ 6 U ^ clc ðAÞ ¼ A. Since A 6 AKb and A 6 clc ðAÞ; A 6 AKb ^ clc ðAÞ. (c) ) (a): Since AKb is a fuzzy Kb set and clc ðAÞ is a fuzzy c-closed set, A is a fuzzy locally Kb set. h 5. Fuzzy Kb continuity and fuzzy locally Kb continuity In this section we present two weaker forms of fuzzy continuity named fuzzy Kb continuity and fuzzy locally Kb continuity via the fuzzy Kb sets and fuzzy locally Kb sets and we obtain some characterizations of these new continuities. Definition 5.1. Let f : ðX; s1 Þ ! ðY; s2 Þ be a function from a fuzzy topological space ðX; s1 Þ into a fuzzy topological space ðY; s2 Þ. The function f is called (a) fuzzy continuous [4] if f 1 ðBÞ is a fuzzy open set of X for each B 2 s2 , (b) fuzzy semicontinuous [2] if f 1 ðBÞ is a fuzzy semiopen set of X for each B 2 s2 , (c) fuzzy precontinuous [6] if f 1 ðBÞ is a fuzzy preopen set of X for each B 2 s2 , (d) fuzzy strongly semicontinuous [7] if f 1 ðBÞ is a fuzzy a-open set of X for each B 2 s2 , (e) fuzzy c-continuous [5] if f 1 ðBÞ is a fuzzy c-open set of X for each B 2 s2 , (f) fuzzy semiprecontinuous [12] if f 1 ðBÞ is a fuzzy semipreopen set of X for each B 2 s2 , (g) fuzzy Kb continuous if f 1 ðBÞ is a fuzzy Kb set of X for each B 2 s2 . (h) fuzzy locally Kb continuous if f 1 ðBÞ is a fuzzy locally Kb set of X for each B 2 s2 .
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Remark 5.2. It is clear from the Remarks 3.8 and 4.2 that the implications of the following diagram hold.
fuzzy
fuzzy
precontinous % fuzzy continuous
!
semiprecontinuous &
%
fuzzy strongly
fuzzy
semicontinuous
c-continuous &
%
&
fuzzy
fuzzy
semicontinuous
Kb continuous # fuzzy locally Kb continuous
However, none of the implications of this diagram is reversed as shown in the following examples. Example 5.3. Let X ¼ fa; bg and the fuzzy topology s on X defined as in Example 3.7(a), Y ¼ f1; 2g; Cð1Þ ¼ 0:1; Cð2Þ ¼ 0:4 and the fuzzy topology on Y be # ¼ f0; 1; Cg. Now consider the map f : ðX; sÞ ! ðY; #Þ, defined as follows: f ðaÞ ¼ 1; f ðbÞ ¼ 2. Since
f 1 ðCÞðaÞ ¼ Cðf ðaÞÞ ¼ Cð1Þ ¼ 0:1; f 1 ðCÞðbÞ ¼ Cðf ðbÞÞ ¼ Cð2Þ ¼ 0:4; as shown in Example 3.7(a), f 1 ðCÞ ¼ B R cOðXÞ, but f 1 ðCÞ ¼ B 2 Kb ðXÞ. f is fuzzy Kb continuous, but not fuzzy c-continuous and hence f is not fuzzy semicontinuous, precontinuous, fuzzy strongly semicontinuous and continuous. Example 5.4. Let X ¼ fa; bg and the fuzzy topology s on X defined as in Example 3.10, and Y ¼ f1; 2g and the fuzzy set C on Y defined as follows: Cð1Þ ¼ 0:1; Cð2Þ ¼ 1. Let the fuzzy topology on Y be # ¼ f0; 1; Cg and the map f : ðX; sÞ ! ðY; #Þ defined as follows: f ðaÞ ¼ 1, f ðbÞ ¼ 2. Since
f 1 ðCÞðaÞ ¼ Cðf ðaÞÞ ¼ Cð1Þ ¼ 0:1; f 1 ðCÞðbÞ ¼ Cðf ðbÞÞ ¼ Cð2Þ ¼ 1; as shown in Example 4.3 f 1 ðCÞ ¼ B is not a fuzzy Kb set but it is a fuzzy locally Kb set. Thus, f is not a fuzzy Kb continuous but it is a fuzzy locally Kb continuous function. Theorem 5.5. Let f : ðX; s1 Þ ! ðY; s2 Þ be a function from a fuzzy topological space ðX; s1 Þ into a fuzzy topological space ðY; s2 Þ. The following statements are equivalent: (a) f is fuzzy Kb continuous. (b) For all Bc 2 s2 ; f 1 ðBÞ 2 V b ðXÞ. (c) For all fuzzy set A of Y; ðf 1 ðintAÞÞKb 6 f 1 ðAÞ. Proof. (a) ) (b): Let Bc 2 s2 . Since f is Kb continuous, f 1 ðBc Þ ¼ ðf 1 ðBÞÞc 2 Kb ðXÞ. Thus, f 1 ðBÞ 2 V b ðXÞ. (b) ) (a): It can be proved in the above manner. (a) ) (c): Since intðAÞ 2 s2 and f is Kb continuous,
ðf 1 ðintAÞÞKb ¼ f 1 ðintAÞ 6 f 1 ðAÞ: (c) ) (a): Let B 2 s2 . Then intB ¼ B. By assumption,
ðf 1 ðBÞÞKb ¼ ðf 1 ðintBÞÞKb 6 f 1 ðBÞ: Since f 1 ðBÞ 6 ðf 1 ðBÞÞKb always holds, f 1 ðBÞ 2 Kb ðXÞ. h Theorem 5.6. Let f : ðX; s1 Þ ! ðY; s2 Þ be a function from a fuzzy topological space ðX; s1 Þ into a fuzzy topological space ðY; s2 Þ. The following statements are equivalent: (a) f is fuzzy locally Kb continuous. V (b) For all fuzzy set A of Y; f 1 ðintAÞ ¼ ðf 1 ðintAÞÞKb clc ðf 1 ðintAÞÞ. Vb W 1 1 int c ðf 1 ðclAÞÞ. (c) For all fuzzy set A of Y; f ðclAÞ ¼ ðf ðclAÞÞ
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Proof. (a) , (b): Since intA is a fuzzy open set, the proof is immediate from Theorem 4.5. (a) ) (c): ðclAÞc is a fuzzy open set, so by Theorem 4.5,
f 1 ððclAÞc Þ ¼ ðf 1 ðclAÞÞc ¼ ððf 1 ðclAÞÞc ÞKb
^
clc ððf 1 ðclAÞÞc Þ ^ ¼ ððf 1 ðclAÞÞ Þ ðintc ðf 1 ðclAÞÞÞc ; Vb c
W f 1 ðclAÞ ¼ ðf 1 ðclAÞÞV b int c ðf 1 ðclAÞÞ. (c) ) (a): Let A be a fuzzy open set. Thus, clðAc Þ ¼ Ac . By assumption,
_ f 1 ðclðAc ÞÞ ¼ f 1 ðAc Þ ¼ ðf 1 ðclðAc ÞÞÞV b intc ðf 1 ðclðAc ÞÞÞ _ c V ¼ ðf 1 ðA ÞÞ b intc ðf 1 ðAc ÞÞ _ ¼ ððf 1 ðAÞÞc ÞV b int c ððf 1 ðAÞÞc Þ _ ¼ ððf 1 ðAÞÞKb Þc ðclc ððf 1 ðAÞÞÞÞc ^ clc ððf 1 ðAÞÞÞÞc : ¼ ððf 1 ðAÞÞKb Hence, f 1 ðAÞ ¼ f 1 ðAÞÞKb
V
clc ððf 1 ðAÞÞÞ which means f 1 ðAÞ is a fuzzy locally Kb set. h
Theorem 5.7. Let f : X ! Y be a fuzzy Kb continuous function and g : Y ! Z fuzzy continuous function. Then g f : X ! Z is a fuzzy Kb continuous function. Proof. It is clear from the equality ðg f Þ1 ðBÞ ¼ f 1 ðg 1 ðBÞÞ.
h
Theorem 5.8. Let f : X ! Y be a fuzzy continuous function. If g : X ! X Y the graph map of f is fuzzy Kb continuous, then f is a fuzzy Kb continuous function. Proof. Let B an open set of Y. Then 1X B is an open set of X Y. By Lemma 2.4. in [2],
g 1 ð1X BÞ ¼ 1X ^ f 1 ðBÞ ¼ f 1 ðBÞ 2 Kb ðXÞ:
6. Conclusion The significance of fuzzy topology has rapidly been appeared in many fields of both pure and applied sciences. The notions of fuzzy continuity and fuzzy open sets are fundamental structures of fuzzy topology. That is why the weaker forms of fuzzy continuity and open sets have been studied by many mathematicians. The weaker types of fuzzy open sets and fuzzy continuity, established in this paper, may play an important role in computational fuzzy topology which was developed in [11] and may have applications in quantum physics, particularly in connection with string theory and 1 theory [8–10]. References Andrijevic´ D. On b-open sets. Mat Vesnik 1996;48:59–64. Azad KK. On fuzzy semicontinuity, fuzzy almost continuity and fuzzy weakly continuity. J Math Anal Appl 1981;82:14–32. Caldas M, Jafari S, Noiri T. On Kb -sets and the associated topology sKb . Acta Math Hung 2006;110(4):337–45. Chang CL. Fuzzy topological spaces. J Math Anal Appl 1968;24:182–90. Hanafy IM. Fuzzy c-open sets and fuzzy c-continuity. J Fuzzy Math 1999;7:419–30. Singal MK, Prakash N. Fuzzy preopen sets and fuzzy preseparation axioms. Fuzzy Sets Syst 1991;44:273–81. Bin Shahna AS. On fuzzy strong semicontinuity and fuzzy precontinuity. Fuzzy Sets Syst 1991;44:303–8. El Naschie MS. On the uncertainty of cantorian geometry and the two-slit experiment. Chaos, Solitons and Fractals 1998;9(3):517–29. El Naschie MS. On the unification of heterotic strings, M theory and 1 theory. Chaos, Solitons and Fractals 2000;11(14):2397–408. El Naschie MS. On a fuzzy Kahler-like manifold which is consistent with the two slit experiment. Int J Nonlinear Sci Numer Simul 2005;6(2):94–7. Shi Wenzhong, Liu Kimfung. A fuzzy topology for computing the interior, boundary, and exterior of spatial objects quantitatively in GIS. Comput Geosci 2007;33:898–915. [12] Thakur SS, Singh S. On fuzzy semi-preopen sets and fuzzy semi-precontinuity. Fuzzy Sets Syst 1998;98:383–91. [13] Zadeh LA. Fuzzy sets. Inform Control 1965;8:338–53. [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]