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R1 fuzzy topological spaces
JOURNAL
OF MATHEMATICAL
ANALYSIS
AND
127, 151-154 (1987)
APPLICATIONS
R, Fuzzy Topological
Spaces
ARUN K. SRIVASTAVA Deparlment
of Mathematics,
Banaras Submitted
Hindu
University,
Varanasi-221005,
India
by L. A. Zadeh
Received April 9, 1984 A fuzzy topological analog of the R, separation axiom of topology is introduced and its appropriateness is established. 6? I987 Academic Press, Inc
The R, separation axiom was first introduced by Yang [lo] who, however, called it the r1 axiom; the current name R, is due to Davis [2] who also observed that R, + r, = T,. We define here a fuzzy topological analog of this axiom and attempt to establish its appropriateness through several observations. It must be mentioned that a fuzzy topological version of the R, axiom has earlier been considered by Hutton and Reilly [3]; however, their approach and point of view appear quite different from ours. The fuzzy topological concepts we make use of are fairly standard by now and can be found, e.g., in Chang [l] and Lowen [IS]. However, unless mentioned to the contrary, all fuzzy topological spaces (fts) considered here, are in Chang’s sense [ 11; specifically, we do not necessarily include all constants in a fuzzy topology as in Lowen [S]. By a fuzzy point in a set X, we mean a fuzzy set in X which has exactly one non-zero value, say r E (0, l), at exactly one point, say x, of X; r and x are respectively the value and support of this fuzzy point denoted by x,. If A is a fuzzy set in X then we write x, E A iff r < A(x). DEFINITION 1. An fts(X, f) is R, iff for all x, YE X, x fy, whenever 3U E t with either U(x) = 1 and U(y) = 0, or U(x) = 0 and U(y) = 1 then for any pair of fuzzy points x, and y,, 3V, WE t with Vn W= a, X,E V, and y, E W.
We first show that our fuzzy R,-ness is a “good extension” of the concept of (topological) R,-ness. For this, recall that a topological space (X, T) is R, iff whenever x, y E X, x # y, and y $ X then 3 disjoint U, V E T with x E U and y E V. If T is a topology on a set X then w(T) stands for the set of all 1.s.c.functions from (X, T) to [0, 11; w(T) is a fuzzy topology on X. THEOREM
1. A topological
space (X, T) is R, iff thefts
(A', o(T))
is R, .
151 0022-247X/87 $3.00 Copyright 0 1987 by Academx Press, Inc. All qhfs of reproduction in any form reserved.
152
ARUN
K.
SRIVASTAVA
Proof First, let (X, T) be R,. Let x, YE X, x#y, be such that 3U E o( T) with U(x) = 1 and U(y) = 0. Consider two fuzzy points x, and y,. As UE T, xu is 1.s.c. and so for a E (0, l), V= 1;’ (a, l] E T. Clearly, x6 V and y$ V and hence x$y. Now as (X, T) is R,, IA, BET such that x E A, y E B, and A n B = @. But then xA and xe, being I.s.c., belong to o(T) and are such that x, E xa, y, E xB, and xa n xe = @. Hence (X, w(T)) is R,. Next, let (X, w(T)) be R, and let x, YE X, x # y, be such that x$y. Then 3lJeT with XEU and y$U. Let V=xu. Then VEO(T), V(x) = 1, and V(y) = 0. Consider two fuzzy points x, and y,. As (X, o(T)) with X,E W,, y,~ W,, and W,n W,=@. Let is R,, 3W,, W,E~(T) U,= W;‘(r, 1J and Us= W;‘(s, 11. Then U,, U,E T, XE U,, and YE Us. Moreover, U, n U, = 0 for if some z E U, n U, then z E W;‘(r, l] n W; ‘(5, l] and so W,(z) > r and W,(z) > s, whence W, n W, cannot be 0, a contradiction. Hence (X, T) is R, .
We next recall the following definitions. DEFINITION 2 ([S, 93). An fts(X, t) is called (i) T,, (ii) R,, and (iii) T, (Hausdorff) according as (i) Vx, YE X, x #y, either HUE t with U(x)= 1 and U(y)=0 or 3V~t with V(x)=0 and V(y)= 1; (ii) VX,~EX, x#y, whenever 3U E t with U(x) = 1 and U(y) = 0, there also exists VE t with V(x) = 0 and V(y) = 1; and (iii) any two fuzzy points of X with distinct
supports are contained in disjoint
t-open fuzzy sets.
We may remark that in [9], a justification for choosing the above definitions of fuzzy T, and R, spaces is presented and a justification for the above definition of fuzzy T, spaces is presented in [8].’ THEOREM
2. For fuzzy topological spaces,R, + T, = T,
Proof: Both “RI + T, =S Ts” and “TZ *R,” are obvious. Now if an fts (X, r) is T, and x, y E X, x # y, consider any two fuzzy points x, and y,. Then 3 disjoint U,, I/,E t with X,E U, and y,~ Vs. Hence r < U,(x) and O=U,(y).LetU=sup{U,:O~r~1}.ThenU~tandclearly,U(x)=1and U(y) = 0. Hence (X, t) is T,. Remark 1. It is easily seen that R, *R, and R, =b R,; for the second non-implication, an obvious modification of the standard counterexample in the topological case, viz., a cofinite topological space, works. are other definitions also. For example Liu and Pu in [4] define fuzzy T,-ness difit turns out to be equivalent to fuzzy T,-ness of Definition 2 above; this is done in Srivastava’s Ph. D thesis 171. where a somewhat comprehensive study of these and similar topics is available. ’ There
ferentlybut
R, FUZZY TOPOLOGICAL
SPACES
153
We now recall from [6] that the initialfuzzy topology on a set X for the family of fts{(Xi, ti)}is, and the family of functions (f;: X-, (Xi, ti)jiE, is the smallest fuzzy topology on X making each f, fuzzy continuous. It is easily seen that it is generated by the family {f,-‘( U,): U, E z;},~ ,. THEOREM 3. The R, property is initial, i.e., if {(Xi, fi)},E, is a family of RI fls and (fi: X+(Xi, fr)}iE,, a family of functions then the initial topology on X for the family { fi} it, is also R 1.
Proof: Let t be the initial topology on A’. Let x, y E A’, x # y, and let HUE t with U(x) = 1 and U(y) = 0. Let x, and y, be two fuzzy points of 2’. For any o!E (0, l), consider the fuzzy point x,. Then x, E U and so it is possible to find a basic fuzzy open set, say f;‘(U:,)nf;YV2)n
Uyk(1
... nfi,‘(V$
being t,-open, such that X,E
inf f,;‘(U;)CU.
(1)
I
But then Va E (0, 1 ), tl<
inf f; ‘(U;)(x)bU(x)
lCk
or
Thus, 1 =SUP I
,f;',,, . .
v&u&H.
Now as Va E (0, 1), q(f&))
G SUP U:k(fr,(x))>
2
therefore, Vtl E (0, 1 ), inf
uY~(fi~(x))
6
l
, f;‘,,
SUP Vk(fifi,(X))
1.
a
and hence, 1 = SUP, ;Iy&
. .
3
q(f&))
d , ,I:[, .’
sup Ul”,(f&)). I
But this implies that sup q(f&)) OL
= 1,
Vk, 1 dkdn.
154
ARUN
K.
SRIVASTAVA
In particular, SUP q(f;,(x)) 2
= 1.
Now let U, = sup, U;,. Then U, E ri, and U,(f,,(x)) = 1. Also, as U(y) = 0, from (l), U;(f.,(y))=O, Vct~(0, 1). Thus U,(fi,(~~))=o. Since (X,,, ti,) is R,, every two fuzzy points of Xi, with supports h,(x) and k.,(y) must be contained in disjoint t,,-open fuzzy sets. Thus &,-open fuzzy sets V, and W, such that (&(x)),E V,, (h,(y)),~ W,, and V, n W, =a. Now let V,=f,7'( V,) and Ws=f,~‘( W,). Then X,E V,, since (fi,(x)),~ V, jr < V,(f,(x)). Similarly, JJ,E W,. Moreover, V, n W, = 0, for if 32~ X with V,(z) > 0 and W,(z) > 0, then V,(z) = f i, ‘( VI)(z) = Vl(fi,(z)) > 0 and similarly, W,(fi,(z)) > 0, contradicting the fact that V, n W, = 0. Hence (X, t) must be R,. A fuzzy topological property which is initial is easily seen to be productive, hereditary (i.e., possessed by each crisp subspace), and projective’ (i.e., when a product fts has it then each factor also possesses it). We close with the following useful corollary of Theorem 3. THEOREM
4.
The fuzzy
R, property
is productive, projective’
and
hereditary. ACKNOWLEDGMENT The author
gratefully
acknowledges
helpful
advice
from
Rekha
Srivastava.
REFERENCES 1. C. L. CHANG, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968), 182-190. 2. A. S. DAMS, Indexed system of neighbourhoods for general topological spaces, Amer. Math. Monthly, 68 (1961), 886-893. 3. B. HUTTON AND I. L. REILLY, Separation axioms in fuzzy topological spaces, Fuzzy Sers and Sysfems 3 (1980), 93-104. 4. Y. M. LIU AND P. M. Pu, Fuzzy topology I, J. Math. Anal. Appl. 76 (1980), 571-599. 5. R. LOWEN, Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl. 56 (1976), 621-633. 6. R. LOWEN, Initial and final fuzzy topologies and the fuzzy Tychonoff theorem, J. Mafh. Anal. Appl. 58 (1977), 11-21. Ph. D thesis, Banaras Hindu University, 7. R. SRIVASTAVA, “Topics in Fuzzy Topology,” India, 1983. 8. R. SRIVASTAVA, S. N. LAL, AND A. K. SRIVASTAVA, Fuuzy Hausdorff topological spaces, J. Math. Anal. Appl. 81 (1981), 497-506. 9. R. SRIVASTAVA, S. N. LAL, AND A. K. SRIVASTAVA, On fuzzy T, and R, topological spaces, J. Math. Anal. Appl., in press. 10. C. T. YANG, On paracompact spaces, Proc. Amer. Math. Sot. 5 (1954), 185-189. 2 Projectivity, however, follows if we restrict to fuzzy those which necessarily contain constants also.
topologies
in Lowen’s
sense [S],
i.e.,
OF MATHEMATICAL
ANALYSIS
AND
127, 151-154 (1987)
APPLICATIONS
R, Fuzzy Topological
Spaces
ARUN K. SRIVASTAVA Deparlment
of Mathematics,
Banaras Submitted
Hindu
University,
Varanasi-221005,
India
by L. A. Zadeh
Received April 9, 1984 A fuzzy topological analog of the R, separation axiom of topology is introduced and its appropriateness is established. 6? I987 Academic Press, Inc
The R, separation axiom was first introduced by Yang [lo] who, however, called it the r1 axiom; the current name R, is due to Davis [2] who also observed that R, + r, = T,. We define here a fuzzy topological analog of this axiom and attempt to establish its appropriateness through several observations. It must be mentioned that a fuzzy topological version of the R, axiom has earlier been considered by Hutton and Reilly [3]; however, their approach and point of view appear quite different from ours. The fuzzy topological concepts we make use of are fairly standard by now and can be found, e.g., in Chang [l] and Lowen [IS]. However, unless mentioned to the contrary, all fuzzy topological spaces (fts) considered here, are in Chang’s sense [ 11; specifically, we do not necessarily include all constants in a fuzzy topology as in Lowen [S]. By a fuzzy point in a set X, we mean a fuzzy set in X which has exactly one non-zero value, say r E (0, l), at exactly one point, say x, of X; r and x are respectively the value and support of this fuzzy point denoted by x,. If A is a fuzzy set in X then we write x, E A iff r < A(x). DEFINITION 1. An fts(X, f) is R, iff for all x, YE X, x fy, whenever 3U E t with either U(x) = 1 and U(y) = 0, or U(x) = 0 and U(y) = 1 then for any pair of fuzzy points x, and y,, 3V, WE t with Vn W= a, X,E V, and y, E W.
We first show that our fuzzy R,-ness is a “good extension” of the concept of (topological) R,-ness. For this, recall that a topological space (X, T) is R, iff whenever x, y E X, x # y, and y $ X then 3 disjoint U, V E T with x E U and y E V. If T is a topology on a set X then w(T) stands for the set of all 1.s.c.functions from (X, T) to [0, 11; w(T) is a fuzzy topology on X. THEOREM
1. A topological
space (X, T) is R, iff thefts
(A', o(T))
is R, .
151 0022-247X/87 $3.00 Copyright 0 1987 by Academx Press, Inc. All qhfs of reproduction in any form reserved.
152
ARUN
K.
SRIVASTAVA
Proof First, let (X, T) be R,. Let x, YE X, x#y, be such that 3U E o( T) with U(x) = 1 and U(y) = 0. Consider two fuzzy points x, and y,. As UE T, xu is 1.s.c. and so for a E (0, l), V= 1;’ (a, l] E T. Clearly, x6 V and y$ V and hence x$y. Now as (X, T) is R,, IA, BET such that x E A, y E B, and A n B = @. But then xA and xe, being I.s.c., belong to o(T) and are such that x, E xa, y, E xB, and xa n xe = @. Hence (X, w(T)) is R,. Next, let (X, w(T)) be R, and let x, YE X, x # y, be such that x$y. Then 3lJeT with XEU and y$U. Let V=xu. Then VEO(T), V(x) = 1, and V(y) = 0. Consider two fuzzy points x, and y,. As (X, o(T)) with X,E W,, y,~ W,, and W,n W,=@. Let is R,, 3W,, W,E~(T) U,= W;‘(r, 1J and Us= W;‘(s, 11. Then U,, U,E T, XE U,, and YE Us. Moreover, U, n U, = 0 for if some z E U, n U, then z E W;‘(r, l] n W; ‘(5, l] and so W,(z) > r and W,(z) > s, whence W, n W, cannot be 0, a contradiction. Hence (X, T) is R, .
We next recall the following definitions. DEFINITION 2 ([S, 93). An fts(X, t) is called (i) T,, (ii) R,, and (iii) T, (Hausdorff) according as (i) Vx, YE X, x #y, either HUE t with U(x)= 1 and U(y)=0 or 3V~t with V(x)=0 and V(y)= 1; (ii) VX,~EX, x#y, whenever 3U E t with U(x) = 1 and U(y) = 0, there also exists VE t with V(x) = 0 and V(y) = 1; and (iii) any two fuzzy points of X with distinct
supports are contained in disjoint
t-open fuzzy sets.
We may remark that in [9], a justification for choosing the above definitions of fuzzy T, and R, spaces is presented and a justification for the above definition of fuzzy T, spaces is presented in [8].’ THEOREM
2. For fuzzy topological spaces,R, + T, = T,
Proof: Both “RI + T, =S Ts” and “TZ *R,” are obvious. Now if an fts (X, r) is T, and x, y E X, x # y, consider any two fuzzy points x, and y,. Then 3 disjoint U,, I/,E t with X,E U, and y,~ Vs. Hence r < U,(x) and O=U,(y).LetU=sup{U,:O~r~1}.ThenU~tandclearly,U(x)=1and U(y) = 0. Hence (X, t) is T,. Remark 1. It is easily seen that R, *R, and R, =b R,; for the second non-implication, an obvious modification of the standard counterexample in the topological case, viz., a cofinite topological space, works. are other definitions also. For example Liu and Pu in [4] define fuzzy T,-ness difit turns out to be equivalent to fuzzy T,-ness of Definition 2 above; this is done in Srivastava’s Ph. D thesis 171. where a somewhat comprehensive study of these and similar topics is available. ’ There
ferentlybut
R, FUZZY TOPOLOGICAL
SPACES
153
We now recall from [6] that the initialfuzzy topology on a set X for the family of fts{(Xi, ti)}is, and the family of functions (f;: X-, (Xi, ti)jiE, is the smallest fuzzy topology on X making each f, fuzzy continuous. It is easily seen that it is generated by the family {f,-‘( U,): U, E z;},~ ,. THEOREM 3. The R, property is initial, i.e., if {(Xi, fi)},E, is a family of RI fls and (fi: X+(Xi, fr)}iE,, a family of functions then the initial topology on X for the family { fi} it, is also R 1.
Proof: Let t be the initial topology on A’. Let x, y E A’, x # y, and let HUE t with U(x) = 1 and U(y) = 0. Let x, and y, be two fuzzy points of 2’. For any o!E (0, l), consider the fuzzy point x,. Then x, E U and so it is possible to find a basic fuzzy open set, say f;‘(U:,)nf;YV2)n
Uyk(1
... nfi,‘(V$
being t,-open, such that X,E
inf f,;‘(U;)CU.
(1)
I
But then Va E (0, 1 ), tl<
inf f; ‘(U;)(x)bU(x)
lCk
or
Thus, 1 =SUP I
,f;',,, . .
v&u&H.
Now as Va E (0, 1), q(f&))
G SUP U:k(fr,(x))>
2
therefore, Vtl E (0, 1 ), inf
uY~(fi~(x))
6
l
, f;‘,,
SUP Vk(fifi,(X))
1.
a
and hence, 1 = SUP, ;Iy&
. .
3
q(f&))
d , ,I:[, .’
sup Ul”,(f&)). I
But this implies that sup q(f&)) OL
= 1,
Vk, 1 dkdn.
154
ARUN
K.
SRIVASTAVA
In particular, SUP q(f;,(x)) 2
= 1.
Now let U, = sup, U;,. Then U, E ri, and U,(f,,(x)) = 1. Also, as U(y) = 0, from (l), U;(f.,(y))=O, Vct~(0, 1). Thus U,(fi,(~~))=o. Since (X,,, ti,) is R,, every two fuzzy points of Xi, with supports h,(x) and k.,(y) must be contained in disjoint t,,-open fuzzy sets. Thus &,-open fuzzy sets V, and W, such that (&(x)),E V,, (h,(y)),~ W,, and V, n W, =a. Now let V,=f,7'( V,) and Ws=f,~‘( W,). Then X,E V,, since (fi,(x)),~ V, jr < V,(f,(x)). Similarly, JJ,E W,. Moreover, V, n W, = 0, for if 32~ X with V,(z) > 0 and W,(z) > 0, then V,(z) = f i, ‘( VI)(z) = Vl(fi,(z)) > 0 and similarly, W,(fi,(z)) > 0, contradicting the fact that V, n W, = 0. Hence (X, t) must be R,. A fuzzy topological property which is initial is easily seen to be productive, hereditary (i.e., possessed by each crisp subspace), and projective’ (i.e., when a product fts has it then each factor also possesses it). We close with the following useful corollary of Theorem 3. THEOREM
4.
The fuzzy
R, property
is productive, projective’
and
hereditary. ACKNOWLEDGMENT The author
gratefully
acknowledges
helpful
advice
from
Rekha
Srivastava.
REFERENCES 1. C. L. CHANG, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968), 182-190. 2. A. S. DAMS, Indexed system of neighbourhoods for general topological spaces, Amer. Math. Monthly, 68 (1961), 886-893. 3. B. HUTTON AND I. L. REILLY, Separation axioms in fuzzy topological spaces, Fuzzy Sers and Sysfems 3 (1980), 93-104. 4. Y. M. LIU AND P. M. Pu, Fuzzy topology I, J. Math. Anal. Appl. 76 (1980), 571-599. 5. R. LOWEN, Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl. 56 (1976), 621-633. 6. R. LOWEN, Initial and final fuzzy topologies and the fuzzy Tychonoff theorem, J. Mafh. Anal. Appl. 58 (1977), 11-21. Ph. D thesis, Banaras Hindu University, 7. R. SRIVASTAVA, “Topics in Fuzzy Topology,” India, 1983. 8. R. SRIVASTAVA, S. N. LAL, AND A. K. SRIVASTAVA, Fuuzy Hausdorff topological spaces, J. Math. Anal. Appl. 81 (1981), 497-506. 9. R. SRIVASTAVA, S. N. LAL, AND A. K. SRIVASTAVA, On fuzzy T, and R, topological spaces, J. Math. Anal. Appl., in press. 10. C. T. YANG, On paracompact spaces, Proc. Amer. Math. Sot. 5 (1954), 185-189. 2 Projectivity, however, follows if we restrict to fuzzy those which necessarily contain constants also.
topologies
in Lowen’s
sense [S],
i.e.,