- Email: [email protected]
On decomposition of proper compactness in fuzzy topological spaces
JOURNAL
OF MATHEMATICAL
ANALYSIS
AND
APPLICATIONS
134,
46.50 (1988)
On Decomposition of Proper Compactness in Fuzzy Topological Spaces JINGCHENG
TONG
Department of Mathematics, Wayne State University, Detroit, MI 48202 U.S.A and btstitute of Applied Mathematics, Academia Sinica, Peking, China Submitted by L. A. Zadeh Received May 17, 1985
In this paper we introduce the notions of 2-compactness and fuzzy metacompactness for fuzzy topological spaces and prove the following theorem: THEOREM. A fuzzy topological space X is properly compact &iJit is 2-compact and fuzzy metacompact. f? 1988 Academic Press, Inc
1.
INTRODUCTION
An ordinary topological space X is said to be metacompact if every open cover of X has a point finite refinement. Arens and Dugundji [2,6] proved that an ordinary topological space is compact iff it is countably compact and metacompact. Their result has been improved by Aquaro [ 11, Au11 [3], Chaber [4], Wicke and Worrell [13], and Worrell and Wicke [15]. For fuzzy topological spaces, we do not find any investigations of a corresponding result in the literatureThe purpose of this paper is to try to provide such a decomposition theorem. 2.
PRELIMINARIES
The notion of compactness in fuzzy topological spaces (fts) wds first introduced by Chang [S]. DEFINITION 1. A fts X has a finite subcover.
is compact
iff every
cover
of X of open fuzzy sets
In the above definition, a cover Q of X is a collection of fuzzy sets A, (iel) in X such that XcUi.,Ai. If the index set I is finite, then Xc Uiel Ai implies that each fuzzy point in X belongs to a certain fuzzy set 46 0022-247X/88 $3.00 Copyright 0 1988 by Academic Press. Inc. All rights of reproduction in any form reserved.
DECOMPOSITION
OF PROPER COMPACTNESS
47
A~, (i, E I). But if I is infinite, XC Uic, A, does not imply that each fuzy point in X belongs to a certain A I,,. This makes the notion of “cover” very inconvenient in the study of fuzzy compactness. The following stronger form of the notion of cover is more appropriate, it as given by Sakar in [lo, 111. DEFINITION 2. Let X be a fts. A collection d&of fuzzy sets A I (i E I) in X is said to be a proper cover if for each fuzzy point x in X, there is a fuzzy set A,E% (&EZ) such that SEA,. DEFINITION 3. A fts X is said to be properly compact if each proper cover of X of open fuzzy sets has a finite subcover. DEFINITION 4. A fts X is said to be properly countably compact if each proper cover of X of countably many open fuzzy sets has a finite subcover. DEFINITION 5. Let d be a set of fuzzy points in a fts X. If each open fuzzy set containing a fuzzy point x contains at least m (or infinite) fuzzy points in d, then the fuzzy point x is said to be an m- (w-) limit point of .&z. DEFINITION 6. A fts X is said to be m- (w-) compact if every set .d of infinitely many fuzzy points in X has an m- (CD-)limit point.
It is easily seen that if m 2 n, then an m-compact fts X is also n-compact; an w-compact fts X is m-compact for all positive integers m. EXAMPLE 1. 2-compactness does not imply proper countable compactness. Let X = [0, 11, and r,, be a sequenceof rational numbers in (0, 1) such that O
if u < r,,, otherwise; if u is irrational or u = I, otherwise. Let t be the fuzzy topology generated by ~1, (n = 1,2, ...) and v. Then (X, z) is not properly countably compact since { pn, v} is a proper cover of X of countably many open fuzzy sets without a finite subcover. Now we prove that (X, r) is 2-compact. Let ZZIbe a set of infinitely many fuzzy points in X. If there are two fuzzy points in d having the same sup-
48
JINGCHENG TONG
port 1, then the fuzzy point with support 1 and value 1 is a 2-limit point of d. If there are two fuzzy points in d having supports t, and tz such that maX(t,, tz) < 1, then let n=min{k; rkamax(ti, t,)}. It is easily seen that every fuzzy point with rational support such that rn -Csupp x 6 r, + 1 and value 1 at supp x is a 24imit point of d. Therefore (X, t) is a 2-compact space. DEFINITION 7. Let %!= ( Ui; i E Z} be a proper cover of a fts X of open fuzzy sets. If V = { Vj;j~ .Z} is another proper cover of X of open fuzzy sets, and for each Vj in Y, there is a Ui in % such that Vjc U;, then V is said to be a refinement of %!. DEFINITION 8. Let 4 be a proper cover of a fts X of open fuzzy sets. A fuzzy point x in X is said to be a a’. f point if the number of all open fuzzy sets in % containing x is finite. DEFINITION 9. A fts X is said to be fuzzy metacompact if each proper cover 9 of X of open fuzzy sets has a refinement ^Y-such that the union of the supports of all Y.f points is the set X.
Let X be a fuzy metacompact fts. Let %! be a proper cover of X of open fuzzy sets, Y be its refinement such that the union of the supports of all Y.f points is the set X. Zf 0 is an open fuzzy set different from the fuzzy set .I’, then there is a Y.f point x such that x 4 0. LEMMA 1.
Since 0 is different from the fuzzy set X, there is a fuzzy point y such that y .#0. Let z be a Y. f point with supp z = supp y. If the value of y at supp y is less than the value of z at supp z, then since y # 0, we know ~$0. If the value of y at supp y is larger than the value of z at supp z, then since z belongs to finitely many open fuzzy sets in “Y, y belongs to finitely many open fuzzy sets in Y also, i.e., y is a Y.f point. It is known that y $0. Therefore in either case we can find a V .f point x # 0. ProoJ:
3. A DECOMPOSITION THEOREM OF PROPER COMPACTNESS THEOREM
1. A fts X is properly compact iff it is 2-compact and fuzzy
metacompact. Prooj: Necessity. It is trivial that a properly compact fts X is fuzzy metacompact, we need only to show that a properly compact fts X is 2-compact. In fact we can prove a stronger assertion that a properly compact fts X is o-compact.
DECOMPOSITION
OF PROPER COMPACTNESS
49
Let d be a set of infinitely many fuzzy points in X without o-limit point. Consider an arbitrary fuzzy point x in X. Since x is not an w-limit point Of d, there is an open fuzzy set 0, such that x E 0 ‘i and 0 ~ contains at most finitely many fuzzy points of d. It is easily seen that (0,; x E X} is a proper cover of X of open fuzzy sets. Hence there is a positive integer m such that Xc U:=, 0,. But UT=, O.,, contains at most finitely many fuzzy points of d, and each fuzzy point in A? belongs to a certain 0 ri, hence the set ZZ’is finite, a contradiction. Sufficiency. We prove that if a fts X is fuzzy metacompact but X is not properly compact, then X is not 2-compact. Let @ be a proper cover of X of open fuzzy sets which has a finite subcover. Let V be the refinement of &, for which the union of the supports of all Y‘.f points is the set X. Choose a V.f point x, in X. Since A//‘ is a proper cover, there is an 0, E Y such that x, E 0,. Let 0, be the collection of all the open fuzzy sets in V” containing x, . Then 0, is a finite set. Denote the union of all open fuzzy sets in 0, by V 0,) then V 0, # X. By Lemma 1, there is a V.f point x2 $ V 0,. Since V is a proper cover, there is an O2 E %“^such that X*E 0,. It is easily seen that Oze V\O,, otherwise iz E V fi, . Let 4 be the collection of all the open fuzzy sets in v \ fi, containing x2. Then o2 is’a finite set. Denote the union of all open fuzzy sets in 0,u0* by V(0,uU2), then V(O,u0*)#X. By Lemma 1, there is a %‘.f point x3 $ V (0, u 4). Since VF is a proper cover, there is an 03 E V/“\ (8, u 4) such that x3 E 0, E V\(B, u c2). Inductively we can find a sequence of V“.f points x, E 0, E V\ Ur= 11&, such that the set o,, of all open fuzzy sets in Y\U::: 0, containing x, is a finite set, and if V (U::-,’ I-“,) denotes the union of all open fuzzy sets in IJ;:,’ G;, then -%4V (U721’ q”,). NOW we prove the following relation:
If m > n, then x, $ V (UyZP,’Q), hence x, 4 Q. If m < n, then for each 0, E U,,,,we have x, E 0,~ V”\U::,’ Q, hence 0, $ U?:; Q, and x, $ O,,, x, $ V 0”. Therefore V 0” n Vi”=, xi = x,. Now we prove that the set of fuzzy points d = (xn; n = 1,2, ...} has no 2-limit points. Consider an arbitrary fuzzy point x in X. Since V is a proper cover of X of open fuzzy sets, there is an open fuzzy set 0, in Y’ such that x E 0,. If Ox+ Up”=, Q, then 0, is an open fuzzy set containing x which contains no fuzzy points of d. If 0, E U ,“=, 0,) then 0, E oiOfor certain i,. Since 0, c V Co,and V Q,,n Vp”=, x, = Xi,,, we know that the open fuzzy set 0 ‘i containing x contains just one point x;,, of the set d. Therefore we have proved that if a fts X is fuzzy metacompact but X is not properly
50
JINGCHENG TONG
compact, then it is not 2-compact. This proves that a 2-compact and fuzzy metacompact fts X is properly compact.
ACKNOWLEDGMENT The author wishes to express his thankfulness to Professor T. Nishiura for his valuable help.
REFERENCES 1. G. AQUARO, Point countable open covering in countably compact space, in “General Topology and Its Relation to Modern Analysis and Algebra II, Proceedings, 2nd Prague
Topology Sympos., 1966,” pp. 3941, Acad. Publ. House, Czech. Acad. Sci., 1966. Remarks on the concept of compactness, Portugal Math. 9 (1950) 141-143. 3. C. E. AULL, A generalization of a theorem of Aquaro, Bull. Austral. Mafh. Sot. 9 (1973), 105-108. 4. J. CHABER, Conditions which imply compactness in countably compact spaces,Bull. Acad. Polon. Sci. Ser. Sri. Math. Astronom. Phy. 24 (1976), 993-998. 5. C. L. CHANG, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968), 182-190. 6. J. DUGUNDJI, “Topology,” Allyn & Bacon, 1966. 7. T. E. GANTNER, R. C. STEINLACE, AND R. H. WARREN, Compactness in fuzzy topological spaces, J. Math. Anal. Appl. 62 (1978), 547-562. 8. R. LOWEN, Fuzzy topological spaces and fuzzy compactness, J. Marh. Anal. Appl. 56 (1976), 621-633. 9. R. LOWEN, A comparison of different compactness notions in fuzzy topological spaces, J. Math. nal. Appl. 64 (1978), 446454. 10. M. SAKAR, On fuzzy topological spaces, J. Math. Anal Appl. 79 (1981), 384394. 11. M. SAKAR, On L-fuzzy topological spaces, J. Math. Anal. Appl. 84 (1981) 431442. 12. G. WANG, A new fuzzy compactness defined by fuzzy nets, J. Marh. Anal. Appl. 94 (1983). 1-23. 13. H. H. WICKE AND J. M. WORRELL, JR., Point countability and compactness, Proc. Amer. Math. Sot. 55 (1976), 427431. 14. C. K. WONG, Covering properties of fuzzy topological spaces, J. Math. Anal. Appl. 43 (1973), 697-704. 15. J. M. WORRELL, JR. AND H. H. WICKE, Characterizations of developable topological spaces, Canad. J. Mafh. 17 (1965), 820-830. 16. L. A. ZADEH, Fuzzy sets, Inform. Control 8 (1965), 338-353. 2. R. ARENS AND J. DUGUNDJI,
OF MATHEMATICAL
ANALYSIS
AND
APPLICATIONS
134,
46.50 (1988)
On Decomposition of Proper Compactness in Fuzzy Topological Spaces JINGCHENG
TONG
Department of Mathematics, Wayne State University, Detroit, MI 48202 U.S.A and btstitute of Applied Mathematics, Academia Sinica, Peking, China Submitted by L. A. Zadeh Received May 17, 1985
In this paper we introduce the notions of 2-compactness and fuzzy metacompactness for fuzzy topological spaces and prove the following theorem: THEOREM. A fuzzy topological space X is properly compact &iJit is 2-compact and fuzzy metacompact. f? 1988 Academic Press, Inc
1.
INTRODUCTION
An ordinary topological space X is said to be metacompact if every open cover of X has a point finite refinement. Arens and Dugundji [2,6] proved that an ordinary topological space is compact iff it is countably compact and metacompact. Their result has been improved by Aquaro [ 11, Au11 [3], Chaber [4], Wicke and Worrell [13], and Worrell and Wicke [15]. For fuzzy topological spaces, we do not find any investigations of a corresponding result in the literatureThe purpose of this paper is to try to provide such a decomposition theorem. 2.
PRELIMINARIES
The notion of compactness in fuzzy topological spaces (fts) wds first introduced by Chang [S]. DEFINITION 1. A fts X has a finite subcover.
is compact
iff every
cover
of X of open fuzzy sets
In the above definition, a cover Q of X is a collection of fuzzy sets A, (iel) in X such that XcUi.,Ai. If the index set I is finite, then Xc Uiel Ai implies that each fuzzy point in X belongs to a certain fuzzy set 46 0022-247X/88 $3.00 Copyright 0 1988 by Academic Press. Inc. All rights of reproduction in any form reserved.
DECOMPOSITION
OF PROPER COMPACTNESS
47
A~, (i, E I). But if I is infinite, XC Uic, A, does not imply that each fuzy point in X belongs to a certain A I,,. This makes the notion of “cover” very inconvenient in the study of fuzzy compactness. The following stronger form of the notion of cover is more appropriate, it as given by Sakar in [lo, 111. DEFINITION 2. Let X be a fts. A collection d&of fuzzy sets A I (i E I) in X is said to be a proper cover if for each fuzzy point x in X, there is a fuzzy set A,E% (&EZ) such that SEA,. DEFINITION 3. A fts X is said to be properly compact if each proper cover of X of open fuzzy sets has a finite subcover. DEFINITION 4. A fts X is said to be properly countably compact if each proper cover of X of countably many open fuzzy sets has a finite subcover. DEFINITION 5. Let d be a set of fuzzy points in a fts X. If each open fuzzy set containing a fuzzy point x contains at least m (or infinite) fuzzy points in d, then the fuzzy point x is said to be an m- (w-) limit point of .&z. DEFINITION 6. A fts X is said to be m- (w-) compact if every set .d of infinitely many fuzzy points in X has an m- (CD-)limit point.
It is easily seen that if m 2 n, then an m-compact fts X is also n-compact; an w-compact fts X is m-compact for all positive integers m. EXAMPLE 1. 2-compactness does not imply proper countable compactness. Let X = [0, 11, and r,, be a sequenceof rational numbers in (0, 1) such that O
if u < r,,, otherwise; if u is irrational or u = I, otherwise. Let t be the fuzzy topology generated by ~1, (n = 1,2, ...) and v. Then (X, z) is not properly countably compact since { pn, v} is a proper cover of X of countably many open fuzzy sets without a finite subcover. Now we prove that (X, r) is 2-compact. Let ZZIbe a set of infinitely many fuzzy points in X. If there are two fuzzy points in d having the same sup-
48
JINGCHENG TONG
port 1, then the fuzzy point with support 1 and value 1 is a 2-limit point of d. If there are two fuzzy points in d having supports t, and tz such that maX(t,, tz) < 1, then let n=min{k; rkamax(ti, t,)}. It is easily seen that every fuzzy point with rational support such that rn -Csupp x 6 r, + 1 and value 1 at supp x is a 24imit point of d. Therefore (X, t) is a 2-compact space. DEFINITION 7. Let %!= ( Ui; i E Z} be a proper cover of a fts X of open fuzzy sets. If V = { Vj;j~ .Z} is another proper cover of X of open fuzzy sets, and for each Vj in Y, there is a Ui in % such that Vjc U;, then V is said to be a refinement of %!. DEFINITION 8. Let 4 be a proper cover of a fts X of open fuzzy sets. A fuzzy point x in X is said to be a a’. f point if the number of all open fuzzy sets in % containing x is finite. DEFINITION 9. A fts X is said to be fuzzy metacompact if each proper cover 9 of X of open fuzzy sets has a refinement ^Y-such that the union of the supports of all Y.f points is the set X.
Let X be a fuzy metacompact fts. Let %! be a proper cover of X of open fuzzy sets, Y be its refinement such that the union of the supports of all Y.f points is the set X. Zf 0 is an open fuzzy set different from the fuzzy set .I’, then there is a Y.f point x such that x 4 0. LEMMA 1.
Since 0 is different from the fuzzy set X, there is a fuzzy point y such that y .#0. Let z be a Y. f point with supp z = supp y. If the value of y at supp y is less than the value of z at supp z, then since y # 0, we know ~$0. If the value of y at supp y is larger than the value of z at supp z, then since z belongs to finitely many open fuzzy sets in “Y, y belongs to finitely many open fuzzy sets in Y also, i.e., y is a Y.f point. It is known that y $0. Therefore in either case we can find a V .f point x # 0. ProoJ:
3. A DECOMPOSITION THEOREM OF PROPER COMPACTNESS THEOREM
1. A fts X is properly compact iff it is 2-compact and fuzzy
metacompact. Prooj: Necessity. It is trivial that a properly compact fts X is fuzzy metacompact, we need only to show that a properly compact fts X is 2-compact. In fact we can prove a stronger assertion that a properly compact fts X is o-compact.
DECOMPOSITION
OF PROPER COMPACTNESS
49
Let d be a set of infinitely many fuzzy points in X without o-limit point. Consider an arbitrary fuzzy point x in X. Since x is not an w-limit point Of d, there is an open fuzzy set 0, such that x E 0 ‘i and 0 ~ contains at most finitely many fuzzy points of d. It is easily seen that (0,; x E X} is a proper cover of X of open fuzzy sets. Hence there is a positive integer m such that Xc U:=, 0,. But UT=, O.,, contains at most finitely many fuzzy points of d, and each fuzzy point in A? belongs to a certain 0 ri, hence the set ZZ’is finite, a contradiction. Sufficiency. We prove that if a fts X is fuzzy metacompact but X is not properly compact, then X is not 2-compact. Let @ be a proper cover of X of open fuzzy sets which has a finite subcover. Let V be the refinement of &, for which the union of the supports of all Y‘.f points is the set X. Choose a V.f point x, in X. Since A//‘ is a proper cover, there is an 0, E Y such that x, E 0,. Let 0, be the collection of all the open fuzzy sets in V” containing x, . Then 0, is a finite set. Denote the union of all open fuzzy sets in 0, by V 0,) then V 0, # X. By Lemma 1, there is a V.f point x2 $ V 0,. Since V is a proper cover, there is an O2 E %“^such that X*E 0,. It is easily seen that Oze V\O,, otherwise iz E V fi, . Let 4 be the collection of all the open fuzzy sets in v \ fi, containing x2. Then o2 is’a finite set. Denote the union of all open fuzzy sets in 0,u0* by V(0,uU2), then V(O,u0*)#X. By Lemma 1, there is a %‘.f point x3 $ V (0, u 4). Since VF is a proper cover, there is an 03 E V/“\ (8, u 4) such that x3 E 0, E V\(B, u c2). Inductively we can find a sequence of V“.f points x, E 0, E V\ Ur= 11&, such that the set o,, of all open fuzzy sets in Y\U::: 0, containing x, is a finite set, and if V (U::-,’ I-“,) denotes the union of all open fuzzy sets in IJ;:,’ G;, then -%4V (U721’ q”,). NOW we prove the following relation:
If m > n, then x, $ V (UyZP,’Q), hence x, 4 Q. If m < n, then for each 0, E U,,,,we have x, E 0,~ V”\U::,’ Q, hence 0, $ U?:; Q, and x, $ O,,, x, $ V 0”. Therefore V 0” n Vi”=, xi = x,. Now we prove that the set of fuzzy points d = (xn; n = 1,2, ...} has no 2-limit points. Consider an arbitrary fuzzy point x in X. Since V is a proper cover of X of open fuzzy sets, there is an open fuzzy set 0, in Y’ such that x E 0,. If Ox+ Up”=, Q, then 0, is an open fuzzy set containing x which contains no fuzzy points of d. If 0, E U ,“=, 0,) then 0, E oiOfor certain i,. Since 0, c V Co,and V Q,,n Vp”=, x, = Xi,,, we know that the open fuzzy set 0 ‘i containing x contains just one point x;,, of the set d. Therefore we have proved that if a fts X is fuzzy metacompact but X is not properly
50
JINGCHENG TONG
compact, then it is not 2-compact. This proves that a 2-compact and fuzzy metacompact fts X is properly compact.
ACKNOWLEDGMENT The author wishes to express his thankfulness to Professor T. Nishiura for his valuable help.
REFERENCES 1. G. AQUARO, Point countable open covering in countably compact space, in “General Topology and Its Relation to Modern Analysis and Algebra II, Proceedings, 2nd Prague
Topology Sympos., 1966,” pp. 3941, Acad. Publ. House, Czech. Acad. Sci., 1966. Remarks on the concept of compactness, Portugal Math. 9 (1950) 141-143. 3. C. E. AULL, A generalization of a theorem of Aquaro, Bull. Austral. Mafh. Sot. 9 (1973), 105-108. 4. J. CHABER, Conditions which imply compactness in countably compact spaces,Bull. Acad. Polon. Sci. Ser. Sri. Math. Astronom. Phy. 24 (1976), 993-998. 5. C. L. CHANG, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968), 182-190. 6. J. DUGUNDJI, “Topology,” Allyn & Bacon, 1966. 7. T. E. GANTNER, R. C. STEINLACE, AND R. H. WARREN, Compactness in fuzzy topological spaces, J. Math. Anal. Appl. 62 (1978), 547-562. 8. R. LOWEN, Fuzzy topological spaces and fuzzy compactness, J. Marh. Anal. Appl. 56 (1976), 621-633. 9. R. LOWEN, A comparison of different compactness notions in fuzzy topological spaces, J. Math. nal. Appl. 64 (1978), 446454. 10. M. SAKAR, On fuzzy topological spaces, J. Math. Anal Appl. 79 (1981), 384394. 11. M. SAKAR, On L-fuzzy topological spaces, J. Math. Anal. Appl. 84 (1981) 431442. 12. G. WANG, A new fuzzy compactness defined by fuzzy nets, J. Marh. Anal. Appl. 94 (1983). 1-23. 13. H. H. WICKE AND J. M. WORRELL, JR., Point countability and compactness, Proc. Amer. Math. Sot. 55 (1976), 427431. 14. C. K. WONG, Covering properties of fuzzy topological spaces, J. Math. Anal. Appl. 43 (1973), 697-704. 15. J. M. WORRELL, JR. AND H. H. WICKE, Characterizations of developable topological spaces, Canad. J. Mafh. 17 (1965), 820-830. 16. L. A. ZADEH, Fuzzy sets, Inform. Control 8 (1965), 338-353. 2. R. ARENS AND J. DUGUNDJI,