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Fuzzy points and local properties of fuzzy topological spaces
Fuzzy Sets and Systems 5 (1981) 199-201 • North-Holland Publishing Company
F U Z Z Y P O I N T S A N D L O C A L P R O P E R T I E S OF F U Z Z Y TOPOLOGICAL SPACES Siegfried G O ' I q ' W A L D
Karl-Marx-Universitiit, 701 Leipzig, German Democratic Republic Received October 1979 Revised December 1979 We prove that three theorems of C.K. Wong on local properties of fuzzy topology are wrong, thereby we discuss his notions of fuzzy point, Cp separability, and local compactness.
Keywords: Fuzzy topology, Fuzzy point. Fuzzy points in the sense of Wong [3, 4] are fuzzy singletons in the sense of [1, 5]. 1 However, in contrast to the ideas behind [1, 5], Wong's point of view is to substitute fuzzy points for the usual points of classical topological spaces. Hence, fuzzy membership and fuzzy inclusion are not the right notions for him, he has to define another membership relation: that of fuzzy points with respect to fuzzy sets. Let X be a set with at least two different elements. For A a fuzzy set in X, let us write xeA instead of/xA (x). Then. if [Xo], is the fuzzy point with support xo and value t, we have by [3]
xe[xo]={O
for x = x,,, otherwise,
( 1)
for each x ~ X. Now, Wong's main definition for every fuzzy point p and fuzzy set A in X is:
p~A
iff
[xep]<[xeA]
for all x ~ X .
(2)
Hence, p c A can be true only in case the support Iml of A is the whole of X. As a consequence, for Pl, P2 any fuzzy points, Pl • P2 is always false; a result that corresponds to the facts in case of classical topology. The notions of fuzzy point and boldface membership will not reduce to classical notions in case the generalized membership values are restricted to {0, 1}. Wong's main motivation for these uefinitions therefore is their usefulness in further developments. But, unfortunately, three of his theorems in [3] (or also in [4]) are false. 2 These are Theorems 3.1, 3.2, 3.4 of [3] (i.e. Theorems 5.1, 5.2, 5.4 of [4]). The fuzzy topological notions to be used in the sequel can be found in [3, 4]. The following theorem proves the above m e , t i o n e d theorems to be false. i In [1] the German language equivalent of 'fuzzy singleton' was 't-Einermenge'. 2 It was a student of mine, Brigitte Seidel, who first realized that the proof of Theorem 3.1 of [3] is not correct. 199
S. Gottwald
200
Theorem 1. (i) There exists a family (A~)~ of fuzzy sets and a fuzzy point p, such that p ~ ~,~ A~, but p • A~ is false for every i ~ L (ii) There exists a fuzzy topology if, a base ~ of ~-, an open fuzzy set A of if, and a fuzzy point p~. A, such that p ~ B is false for every member B of ~ . (iii) There exists a fuzzy topological space which is Cn but not separable. l~oof. Choose Xt as the real interval (0, ~r). Let/3 be the set of all open intervals (a, b) with rational endpoints, 0 ~< a, b < w and ] a - b [ < 1. Furthermore, for every subset Z of X let by Z the characteristic function of Z, i.e., Z - X ~ {0, 1} and: Z(x) = 1 iff x ~ Z. Clearly, Z is a fuzzy set in X. Consider the usual topology ~-~ of the subset (0, ~r) of the reals and
Obviously, J~ is a fuzzy topology for X1 and ~ a countable base of ff~. Thus, (X1, ff~) is Cn. But, (X~, ~'t) is not separable: if z =(0.5, 1), then ~ ~ 1 and f:~O (O the empty fuzzy set), but there does not exist any fuzzy point p with p ~ because of 1:214: X1. Hence (iii). To get (ii), choose A =(0, It)^ and P =[1]o.5. Finally, let AI, A2, A 3 , . . . be an enumeration of the elements of ~ . Then A = ~ i ~ N A i and p e A , but p~A~ is false for every i. Hence also (i). To formulate another restriction to the development of local properties of fuzzy topological spaces, let us call in analogy with Lowen [2] a property P' of fuzzy topological spaces a good generalization of a corresponding property P of classical topological spaces iff (a) in case a topological space (X, ~-) has property P, then the fuzzy topological space (X, if), ff = {:~]z E ~-}, has property P' and (b) in case (X, ~-) does not have property It', then also (X, ~') does not have property P'. Otherwise, P' may be called a bad generalization of P. Theorem 2. Each one of the properties
(i) CI, (ii) separability, (iii) local compactness
of fuzzy topological spaces is a bad generalization of the corresponding property of classical topology.
Proof. Consider the space (X1, rl) as in the proof o~ Theorem 1. It is locally compact and separable (in the usual sense). But its corresponding fuzzy version (X1, ~'~) is not separable and also not locally compact in the sense of Wong [3]. For property C~ let be X2 = to2, the initial ordinal of the number class of R2, and or the topology which has as a base the'set of all final segments of to2, or only, all sets {~[~>~ 3~}, 3'
Fuzzy topological spaces
201
At all, not only the notions of fuzzy point and boldface membership are very problematic ones, but also the notions C~, separability, local compactness. A purely formal way o u t w t o save the false 'theorems'---could be another definition of boldface membership:
p~A
iff
[xoep]<[xoeA]
for{xo}=lpl.
(3)
In this case all theorems of 13] would be correct ones. 3 But other difficulties arise; e.g. exists then for every nonempty fuzzy set A in X a family (P~)i~o.~ of fuzzy points such that for all i, j e (0, 1) there hold
(i) Pi ~ A, (ii) piep~ iff i < j . Especially, for each denumerable order type then there exists a descending ~chain of fuzzy points of that order type. Therefore, also (3) seems not to be a much better version than (2). This supports the opinion that one should try to avoid the notion of fuzzy point as defined by Wong altogether for the fuzzification of local properties of fuzzy sets. Perhaps, a way out would be, to use only the concept of 'fuzzy singleton', and to speak of singletons instead of points.
References [1] S. Gottwald, Zahlbereichskonstruktionen in einer mehrwertigen Mengenlehre. Z. Math. Logik Grundl. Math. 17 (1971) 145-188. [2] R. Lowen, A comparison of different compactness notions in fuzzy topological spaces, J. Math. Anal. Appl. 64 (1978) 446--454. [3] C.K. Wong, Fuzzy points and local properties of fuzzy topology, J. Math. Anal. Appl. 46 (19741 316-328. [4] C.K. Wong, Fuzzy topology, in: L.A. Zadeh el al., eds., Fuzzy Sets and Their Applications to Cognitive and Decision Processes (Academic Press, New York, 1975) 171-190. [5] L.A. Zadeh, A fuzzy-set-theoretic interpretation of linguistic hedges, J. Cybernet. 2 (3) (1972) 4-34.
3 This again is ~- result of B. Seidel.
F U Z Z Y P O I N T S A N D L O C A L P R O P E R T I E S OF F U Z Z Y TOPOLOGICAL SPACES Siegfried G O ' I q ' W A L D
Karl-Marx-Universitiit, 701 Leipzig, German Democratic Republic Received October 1979 Revised December 1979 We prove that three theorems of C.K. Wong on local properties of fuzzy topology are wrong, thereby we discuss his notions of fuzzy point, Cp separability, and local compactness.
Keywords: Fuzzy topology, Fuzzy point. Fuzzy points in the sense of Wong [3, 4] are fuzzy singletons in the sense of [1, 5]. 1 However, in contrast to the ideas behind [1, 5], Wong's point of view is to substitute fuzzy points for the usual points of classical topological spaces. Hence, fuzzy membership and fuzzy inclusion are not the right notions for him, he has to define another membership relation: that of fuzzy points with respect to fuzzy sets. Let X be a set with at least two different elements. For A a fuzzy set in X, let us write xeA instead of/xA (x). Then. if [Xo], is the fuzzy point with support xo and value t, we have by [3]
xe[xo]={O
for x = x,,, otherwise,
( 1)
for each x ~ X. Now, Wong's main definition for every fuzzy point p and fuzzy set A in X is:
p~A
iff
[xep]<[xeA]
for all x ~ X .
(2)
Hence, p c A can be true only in case the support Iml of A is the whole of X. As a consequence, for Pl, P2 any fuzzy points, Pl • P2 is always false; a result that corresponds to the facts in case of classical topology. The notions of fuzzy point and boldface membership will not reduce to classical notions in case the generalized membership values are restricted to {0, 1}. Wong's main motivation for these uefinitions therefore is their usefulness in further developments. But, unfortunately, three of his theorems in [3] (or also in [4]) are false. 2 These are Theorems 3.1, 3.2, 3.4 of [3] (i.e. Theorems 5.1, 5.2, 5.4 of [4]). The fuzzy topological notions to be used in the sequel can be found in [3, 4]. The following theorem proves the above m e , t i o n e d theorems to be false. i In [1] the German language equivalent of 'fuzzy singleton' was 't-Einermenge'. 2 It was a student of mine, Brigitte Seidel, who first realized that the proof of Theorem 3.1 of [3] is not correct. 199
S. Gottwald
200
Theorem 1. (i) There exists a family (A~)~ of fuzzy sets and a fuzzy point p, such that p ~ ~,~ A~, but p • A~ is false for every i ~ L (ii) There exists a fuzzy topology if, a base ~ of ~-, an open fuzzy set A of if, and a fuzzy point p~. A, such that p ~ B is false for every member B of ~ . (iii) There exists a fuzzy topological space which is Cn but not separable. l~oof. Choose Xt as the real interval (0, ~r). Let/3 be the set of all open intervals (a, b) with rational endpoints, 0 ~< a, b < w and ] a - b [ < 1. Furthermore, for every subset Z of X let by Z the characteristic function of Z, i.e., Z - X ~ {0, 1} and: Z(x) = 1 iff x ~ Z. Clearly, Z is a fuzzy set in X. Consider the usual topology ~-~ of the subset (0, ~r) of the reals and
Obviously, J~ is a fuzzy topology for X1 and ~ a countable base of ff~. Thus, (X1, ff~) is Cn. But, (X~, ~'t) is not separable: if z =(0.5, 1), then ~ ~ 1 and f:~O (O the empty fuzzy set), but there does not exist any fuzzy point p with p ~ because of 1:214: X1. Hence (iii). To get (ii), choose A =(0, It)^ and P =[1]o.5. Finally, let AI, A2, A 3 , . . . be an enumeration of the elements of ~ . Then A = ~ i ~ N A i and p e A , but p~A~ is false for every i. Hence also (i). To formulate another restriction to the development of local properties of fuzzy topological spaces, let us call in analogy with Lowen [2] a property P' of fuzzy topological spaces a good generalization of a corresponding property P of classical topological spaces iff (a) in case a topological space (X, ~-) has property P, then the fuzzy topological space (X, if), ff = {:~]z E ~-}, has property P' and (b) in case (X, ~-) does not have property It', then also (X, ~') does not have property P'. Otherwise, P' may be called a bad generalization of P. Theorem 2. Each one of the properties
(i) CI, (ii) separability, (iii) local compactness
of fuzzy topological spaces is a bad generalization of the corresponding property of classical topology.
Proof. Consider the space (X1, rl) as in the proof o~ Theorem 1. It is locally compact and separable (in the usual sense). But its corresponding fuzzy version (X1, ~'~) is not separable and also not locally compact in the sense of Wong [3]. For property C~ let be X2 = to2, the initial ordinal of the number class of R2, and or the topology which has as a base the'set of all final segments of to2, or only, all sets {~[~>~ 3~}, 3'
Fuzzy topological spaces
201
At all, not only the notions of fuzzy point and boldface membership are very problematic ones, but also the notions C~, separability, local compactness. A purely formal way o u t w t o save the false 'theorems'---could be another definition of boldface membership:
p~A
iff
[xoep]<[xoeA]
for{xo}=lpl.
(3)
In this case all theorems of 13] would be correct ones. 3 But other difficulties arise; e.g. exists then for every nonempty fuzzy set A in X a family (P~)i~o.~ of fuzzy points such that for all i, j e (0, 1) there hold
(i) Pi ~ A, (ii) piep~ iff i < j . Especially, for each denumerable order type then there exists a descending ~chain of fuzzy points of that order type. Therefore, also (3) seems not to be a much better version than (2). This supports the opinion that one should try to avoid the notion of fuzzy point as defined by Wong altogether for the fuzzification of local properties of fuzzy sets. Perhaps, a way out would be, to use only the concept of 'fuzzy singleton', and to speak of singletons instead of points.
References [1] S. Gottwald, Zahlbereichskonstruktionen in einer mehrwertigen Mengenlehre. Z. Math. Logik Grundl. Math. 17 (1971) 145-188. [2] R. Lowen, A comparison of different compactness notions in fuzzy topological spaces, J. Math. Anal. Appl. 64 (1978) 446--454. [3] C.K. Wong, Fuzzy points and local properties of fuzzy topology, J. Math. Anal. Appl. 46 (19741 316-328. [4] C.K. Wong, Fuzzy topology, in: L.A. Zadeh el al., eds., Fuzzy Sets and Their Applications to Cognitive and Decision Processes (Academic Press, New York, 1975) 171-190. [5] L.A. Zadeh, A fuzzy-set-theoretic interpretation of linguistic hedges, J. Cybernet. 2 (3) (1972) 4-34.
3 This again is ~- result of B. Seidel.