- Email: [email protected]
On fuzzy connectedness
Fuzzy Sets and Systems ~ (1988) 203-208
203
North.Hot|a~ad
ON FUZZY CONN~C~D~S$ Dewan Muslim ALl* and Arun K. SRIVASTAVA D ~ m
of M a ~ ~ ,
~as
Hi~u Umvers~y, Vavas~i221005,India
R_~2e~AvedMay 1986 Revised Janum7 1987 In this note, we discuss some of the properties of various fuzzy connecte~nes~ concepts.
Eeywords:Fuzzytopological space, Fuzzy connectedne~.
Connectedness of a topological space can be described in ~severai equivaient ways, e.g., by (i) non-existence of a proper clopen subset, (fi) non-existence of a proper separation, or (iii) non-existence of a non-constant ccn~nuous function into a particular space (e.g., the two-point discrete space). Most attempts to introduce connectedness in fuzzy topology seem to have been motivated by appropriate translations of either (i) (e.g., Hutton [2], Srivastava [8])or 0i) (e.g., Pu and Liu [5], Rodabaugh [7], Lowen [3], Zheng [10]). In the present note, certain ob~rvat!o " n s pertaining to the connectedness concepts introduced in [2, 5, 7, 8, 10] are made. All undefined concepts and notations we make use of are standard by now (see, e.g. [4]). We, however, particularly mention that the fuzzy °~olmlogies considered here do not necessarily contain constants, mainly because our observations concern works of authors not requiring constants to be fuzzy open. As usual, I denotes the closed unit interval [0,1]. If ~ is a ~ ] ~ a ~ y r set in a set X, ~o MII denote the support of p (i.e., ~offi~-~(0,'l)]. By 1~, we denote the characteristic function of a crisp subset A of X. If t is a fuzzy topology on X, men ~* shall denote the f u r y topology on X generated by t 13 {all constant fuzzy s~t~ in X}; the f ~ y topological space (X, t*) will al~o be denoted as X* wl~e~ no cohesion is possible.
We begin by recalling the following from [5]: two ~ y sets ~ and 7 in X are ca|led Q-separated (rasp. separated.) for a f u ~ topo|o~cal space (X, t) ~ff there exist t-c|osed (rasp. t-open) ~ y sets ~r and ~ in X, such that ~ ~_ ~r, y c ~ an,~ N ~ -- 0 = y N a~. ]t is clear that ~ and y are Q-separated iff ~ N 7 = 0 -- ~ N ~ * Presently at Depamnent of Mathematic, Rsjsh~ Univenity, Rajshahi, Ban~sdesh. 0165-0114/88/$3.50 ~ 1988, E|~v~r Science Pub~sh©~ B.V. (North-Holland)
D. M. A//, A. g. ,~r/wuava ~, it is noted m [5, Prop. 9.2] that if ~ and ~, are crisp then they are ~separated iff~ey are separated. • Several ~ connectedness concepts due to Hutton [2], Lowen [3], P~J and Liu [5], Rodabaugh [7], Srivastava [8] and Zheng [10] exist in the literatu~re; we list them below in the following definition. D e i l d S ~ 2.1. We call a ~ topological space (X, t) (a) FC(i) i~ (X, 0 has no clopen ~ set except 0 and 1 [2]. @) FC(ii) ~ no clopen fuzzy set ~ >>0 (i.e., t~(x)>0, for al~. x) can be (Cl)-separated [31. (c) FC(iii) iff no clopen fuz~ set/~ ~ r > 0 (r ¢ (0, 1]) can be (C2)-separated [31. (d) FC(iv) iff there do not exist non-zero Q-separated fuzzy sets ~ and ? in X = {Sl. (e) FC(v) iff there do not exist non-zero separated fuzzy sets ~ and ? in X with (f) FC(vi) iff there do not exist ~, ~, ¢ t other than 1 and 0 with ~ U ? > 0 ~ d ~nV=0[7]. (g) FC(vh') iff (X, t) has no non-constant clopen f u ~ set [8].
Let IA denote the unit interval with fuz~ topology {0,1,1,4, IA,} where A is some fixed crisp subset of L Let Io denote the unit interval with f~azy topology generated by id and id'. Finally, let ?.D denote the two-point set 2 = {a, b} with topology {0, I, I,, lb}.
~~lou 2.1. A fuzzy topological space (X, t) is FC(i) i~ there exists no fuzzy continuous function f :(X, t)-~ Io with f # 0, f #: 1. Proof. The proof follows from the observation that f is continuous ifff-~(id) and f-1(id') are t-open, or equivalently, ifff is t-elopen. R m ~ 2,1. The fin~ eonnectedness concept FC(i) (due to Hutton [2]) is of no interest if we confine ourselves to fuzzy topologies in ].owen's sense [3] due to the compulsory presence of constants as clopen f u ~ ~sets. In order to repair this aspect of FC(i), Sfivastava [8] modified FC(i) to FC(vii) and obtained its following characterization analogous to Proposition 2.1 above. ~tiou 2~. [8]. A fuzzy topological space (X, t) is FC(vii) iff there exists no non-const~t con~nuous function f :(X, t~-~ I~. R e m ~ 2.2. It is worthwhile to ~ i n t ot~t that the consideration of the concept FC(vii) is somewhat natural in a certain sense. Specifically, I~ has been shown to be canonie~y homeomorphie t~ the space of all probabi~ty measures on the two-point set with discrete topology by Lowen .f4]~ and the two-point discrete space does play the same role in topok~gy as does/~ in Proposition 2.2 above regarding eormeetedness. However, as r~oted in [8], this concept is not a 'good
on.fuzzy connec~ed~.ss
205
extension' (in Lowen's sense [3]) of connectedness sin~ given a topological space (X, T), (X, ~o(T)) is FC(vii) iff there does not exist any non-constant continuous function firom (X, T) to I (with usual topology), a property which connectedness of (X, T) above cannot ~arantee. Our next obsezvation includes a character~mtion of FC(iv) (a~, hence also of FC(v), in View of Proposition 2.4 to follow next). ~~em 2.3. The following statements are equivalem for a fuzzy topological space (X, t):
(a) (x, t) is
cov)
(b) X canno~ be written as the supremum of two non-zero disjoint t-closed fuzzy sets.
(c) X cannot be written as the supremum of two non-zero disjoint t-open fuzzy sets,
(d) The only crisp t-clopen subset of X is 0 or 1. (e) There is no fuzzy continuous function f :(X, ~)--~l~ (or 1.~) other thun 0 and 1.
Proof. (a) :~ (b) :~ (c) :~ (d) is easy. (d) ~ (e) follows from the fact that if f :(X, t)-* G (or 1.~) is f u r y continuous with f ~0, f - 1, then f-1(1~) is proper crisp t-clopen, conU'adicting (d). Finally, to see ( e ) ~ (a), let (X, t) be not FC(iv). Then there exist two non-zero fuzzy sets ~s and y in X with X = ~ U y and f3 7 ffi0 = ~sN ~. Clearly, ~s and y must be crisp and t-clopen. Consider any function f :(X, t)-~/,4 (or I]) such thatf(~) ~_ IA and f(y) ~_ IA.. Then f-1(IA) IA of ----~ and f-t(IA.) = y showing that f is continuous. Evidently, f ~0, f ~ I, contradicting (e). In view of the fact that two crisp subsets are Q-separated iff they are separated, we get the following: l~s|deu
2.4. A fuzzy topological space (X, t) is FC(iv) iff it is FC(v).
We give another characterization of FC(iv) which is sin~lar to the one in Proposition 2.3 ((a) ~ (e)); its easy proof is omitted. P~pos|tton 2.$. A fuzzy topological space (X, t) is FC(iv) iff there is no non-constant fuzzy continuo~ funcffon f "(X, t)--~ 2v (or 2~).
Retook 2.3. The two characterizations of the fuzzy connectedness concept FC(iv) (--FC(v)), v~z. Proposition 2.3 ((a) ¢#(d)) and Proposition 2.5, are nearly identical with similar characterizations of eonnectedness in (usual) topology. All the fuzzy connectedne,,;sconcepts given in defii~ition2.1 are known to be
D. so. At/, A, K. S ~ ~ p r e s e ~ under ~ continuous functions. We now show that some of these ~ m ~ p r ~ r v ~ under some weber f o ~ s of fuzzy continuity alsoo 2.~ ~1]. Given a fuzzy topological space(X, t), a fuzzy set ~ in X is ~ ~ o p e n (rasp. t u g , closed) itt ~ = IntCl/~ (rasp. ~ = CI Int ~). A f ~ c ~ n f:(X, O-e.(y, s) between ~ topologicad spaces is called almost~zy co~nuou~ (resp~ ,.~,eaklyfuzzy c o ~ ~ ) ~ f - l ( l ~ ) e t for each regular open ~ n y ~ t ~ of (y, s) (~esp. ifff "~(/~)~_Intf-~(Cl/0for each/~ ¢ s).
2.6. The fuzzy con~ctedness concepts FC(i), FC(iv), and FC(vfi) are p~erved under both almost and weakly fuzzy continuousfunctions. Proof. We prove the asmrtion only for FC(iv), it being similar for FC(i) and ~(v~). Let f :(X, t)--~ (y, s) be an onto function between fuzzy topological spaces with (y, s)not being FC(iv). Then there exists a proper crisp s-clopen utbset, say/~, in y. Now ~ f is almost fuzzy continuous then as/~ is both regular open and regular c l o ~ , f-a(~) must be proper crisp t-clopen sub~t of X contradicting the fact that (X, t) is FC(~,v). Next, if f is weakly ~ continuous then f-x(/~)=_ Intf-t(Cl~)=IntJ-t(t~) so that f-a(#) is t-open. Similarly f - x ( ~ , ) g Intf-a(Cl ~') = Intf-l(/~ ') so that f-~'(/~') is also t-open. But f-l(/~,) ffi (f-l(~)),. Hence f-t(/,) is a proper crisp t-elopen subset of (X, t), a contradiction. We now give an example which shows that Proposition 2.6 above does not hold good for the remaining fuzzy eonnectedness concepts viz. FC(ii), FC(ifi) end FC(vi). Let Y denote the unit interval I with the fuzzy topology consisting of all constant fuzzy sets. Then Y is clearly FC(ii), FC(iii) and FC(vi). Let X denote the two-point set {a, b) with the fuzzy topology generated by all constant fuzzy sets, and ~ end y, where ~(a) ~ ~ ~ y(b) end/~(b) ---0 - y(a). The constant fuzzy set with value ½is ck~pen in X and (/~, 3') is (C1)- as well as (C2)-separated. So X is neither FC(ii) nor ~(iii). It is evidently not FC(vi) also. Now consider the function f: Y ~ X defined by a i f y E [ o , l], ¢(Y) = # if y E fl, 11.
We assert that y is both ~ o s t and weakly fuzzy continuous. That f is almost fuzzy continuous follows trivially firom the fact that the only regular open fuzzy ~:ets in X are constants. Next, since the ~zz,y ~.~osurei~ X of every open fuzzy set in X is a constant, it is easily verified that f is ~Iso weakly fuzzy continuous. The ~ eonneetedness concepts FC(i) and FC(~i), as noted earlier, are not good extensions of (topological) connectedness. It is shown in [3] that FC(ii) and lFC(i~) are good extensions of conneetedness. Our next result shows that the remarrying fuzzy connectedness concepts, viz, FC(i~) ( = FC(v)) and FC(vi) are ~so good extensions of connectedness.
On ~ z y conneaedness Pm~a or
2.7. A topologicM space (2(, T) is connected iff (X, co(T))~ FC~iv)
FC(,i).
~f. We prove the result only for FC(iv), the proof for FC(vi) being easier. First, let (X, T) be connected. If (X, co(T)) is not FC(iv) then there e~st two non-~ro w(T)-open ~ , ~/~~ y s u c h that Xffi/~U ~, ~ d ~ N y = 0 . Consequently, X ~ o U yo. Also, ~oN 1'o=0, Indeed, if x ~ Yo then ~(x) > 0 so we can choose r e I with y(x) > v > 0 whereby x e y-l(v, 1] G T. Now y-~(v, 1] cannot intersect /Jo, for if x'e~of3y-~(v, 1] then /~(x')>0 and y(x')>r, contradicting the fact that ~ N y = 0. Thus x cannot be a limit point of ~o i.e., x ¢ Po confirming that ~o N Yo= 0. Similarly ~o N ~o = 0. Thus (X, T) is disconnected, a contradiction. The implication (X, co(T)) is FC0v) =~ (X, T) is connected follows from the fact that if A ~ X, then CIe)(T)|A ~ l~r~. We close by presenting the various implications among the fuzzy connectedness concepts FC(i)-FC(vii) in the following proposition. The easy proof is omitted. It is not hard to construct counterexamples to show that no other implications ex~t among these concepts.
~ p o l t i o n 2.8. We have the following implications:
FC(v) FC(i) ~
FC(vii)
~ FC0v) ~
FC(fii)
FC(vO A¢_hew|edgement The authors would like to thank the referee for several pertinent commeat~.
Refe~ ~nces
[1] K.K. Azad, On ~xmy semieontinuity, fuzzy almost ¢ontimaity and fuzzy weakly continuity, J. Math. Anal. Pppl. 82 (1981) 14-32. [2] B. Hutton, Products ot fuzzy toxicologicalspaces, Topology Appl. 11 (1980) 59-67. [3] R. Lowen, Connectedness in ~ topological spaces, Rocky Mount~n J. Math. 11 (1981) 427-433. [4] R. Low©n, On the Existence of Natural Non.topological F ~ Topolo~cal Spaces (Helden~ann Ver|ag, ~t"l~, 1985). [5] P. M. Pu and Y. M. Liu, FuT~v topology l: Neighbourhood structure of a f ~ y pohlt e~nd Moore-Smith convergence, J. M~h. Anal. Appl. 76(1980) 571-599.
D. M. AU, A. K. Sr/vastava
[e] P, M.eu and Y. M. uu, Fuzzy topo~y n, en~u~ and quoaent spa~, ~. Ma~. An~i Appl' 77 098O) 20-3~. [7] S.E. Rcdabaugh, Connectivity and L-fuzzy ~ t interval, Rocky Mountain J. Math. 12 (1982) 113--121. [8] R . Srivutava, Topics in fuzzy topology, Ph.D. Thesis, B~aras Hindu University, V~mesi [9] R. H. Werren, Convergence in fuzzy topology, Rocky Mountain J. Math. 13 (1983)31-36, [to] c. Y. Zbeng, ~zzy path and fuzzy connecte~eu, ~ Sets and Syste~ 14 0984)273-280.
203
North.Hot|a~ad
ON FUZZY CONN~C~D~S$ Dewan Muslim ALl* and Arun K. SRIVASTAVA D ~ m
of M a ~ ~ ,
~as
Hi~u Umvers~y, Vavas~i221005,India
R_~2e~AvedMay 1986 Revised Janum7 1987 In this note, we discuss some of the properties of various fuzzy connecte~nes~ concepts.
Eeywords:Fuzzytopological space, Fuzzy connectedne~.
Connectedness of a topological space can be described in ~severai equivaient ways, e.g., by (i) non-existence of a proper clopen subset, (fi) non-existence of a proper separation, or (iii) non-existence of a non-constant ccn~nuous function into a particular space (e.g., the two-point discrete space). Most attempts to introduce connectedness in fuzzy topology seem to have been motivated by appropriate translations of either (i) (e.g., Hutton [2], Srivastava [8])or 0i) (e.g., Pu and Liu [5], Rodabaugh [7], Lowen [3], Zheng [10]). In the present note, certain ob~rvat!o " n s pertaining to the connectedness concepts introduced in [2, 5, 7, 8, 10] are made. All undefined concepts and notations we make use of are standard by now (see, e.g. [4]). We, however, particularly mention that the fuzzy °~olmlogies considered here do not necessarily contain constants, mainly because our observations concern works of authors not requiring constants to be fuzzy open. As usual, I denotes the closed unit interval [0,1]. If ~ is a ~ ] ~ a ~ y r set in a set X, ~o MII denote the support of p (i.e., ~offi~-~(0,'l)]. By 1~, we denote the characteristic function of a crisp subset A of X. If t is a fuzzy topology on X, men ~* shall denote the f u r y topology on X generated by t 13 {all constant fuzzy s~t~ in X}; the f ~ y topological space (X, t*) will al~o be denoted as X* wl~e~ no cohesion is possible.
We begin by recalling the following from [5]: two ~ y sets ~ and 7 in X are ca|led Q-separated (rasp. separated.) for a f u ~ topo|o~cal space (X, t) ~ff there exist t-c|osed (rasp. t-open) ~ y sets ~r and ~ in X, such that ~ ~_ ~r, y c ~ an,~ N ~ -- 0 = y N a~. ]t is clear that ~ and y are Q-separated iff ~ N 7 = 0 -- ~ N ~ * Presently at Depamnent of Mathematic, Rsjsh~ Univenity, Rajshahi, Ban~sdesh. 0165-0114/88/$3.50 ~ 1988, E|~v~r Science Pub~sh©~ B.V. (North-Holland)
D. M. A//, A. g. ,~r/wuava ~, it is noted m [5, Prop. 9.2] that if ~ and ~, are crisp then they are ~separated iff~ey are separated. • Several ~ connectedness concepts due to Hutton [2], Lowen [3], P~J and Liu [5], Rodabaugh [7], Srivastava [8] and Zheng [10] exist in the literatu~re; we list them below in the following definition. D e i l d S ~ 2.1. We call a ~ topological space (X, t) (a) FC(i) i~ (X, 0 has no clopen ~ set except 0 and 1 [2]. @) FC(ii) ~ no clopen fuzzy set ~ >>0 (i.e., t~(x)>0, for al~. x) can be (Cl)-separated [31. (c) FC(iii) iff no clopen fuz~ set/~ ~ r > 0 (r ¢ (0, 1]) can be (C2)-separated [31. (d) FC(iv) iff there do not exist non-zero Q-separated fuzzy sets ~ and ? in X = {Sl. (e) FC(v) iff there do not exist non-zero separated fuzzy sets ~ and ? in X with (f) FC(vi) iff there do not exist ~, ~, ¢ t other than 1 and 0 with ~ U ? > 0 ~ d ~nV=0[7]. (g) FC(vh') iff (X, t) has no non-constant clopen f u ~ set [8].
Let IA denote the unit interval with fuz~ topology {0,1,1,4, IA,} where A is some fixed crisp subset of L Let Io denote the unit interval with f~azy topology generated by id and id'. Finally, let ?.D denote the two-point set 2 = {a, b} with topology {0, I, I,, lb}.
~~lou 2.1. A fuzzy topological space (X, t) is FC(i) i~ there exists no fuzzy continuous function f :(X, t)-~ Io with f # 0, f #: 1. Proof. The proof follows from the observation that f is continuous ifff-~(id) and f-1(id') are t-open, or equivalently, ifff is t-elopen. R m ~ 2,1. The fin~ eonnectedness concept FC(i) (due to Hutton [2]) is of no interest if we confine ourselves to fuzzy topologies in ].owen's sense [3] due to the compulsory presence of constants as clopen f u ~ ~sets. In order to repair this aspect of FC(i), Sfivastava [8] modified FC(i) to FC(vii) and obtained its following characterization analogous to Proposition 2.1 above. ~tiou 2~. [8]. A fuzzy topological space (X, t) is FC(vii) iff there exists no non-const~t con~nuous function f :(X, t~-~ I~. R e m ~ 2.2. It is worthwhile to ~ i n t ot~t that the consideration of the concept FC(vii) is somewhat natural in a certain sense. Specifically, I~ has been shown to be canonie~y homeomorphie t~ the space of all probabi~ty measures on the two-point set with discrete topology by Lowen .f4]~ and the two-point discrete space does play the same role in topok~gy as does/~ in Proposition 2.2 above regarding eormeetedness. However, as r~oted in [8], this concept is not a 'good
on.fuzzy connec~ed~.ss
205
extension' (in Lowen's sense [3]) of connectedness sin~ given a topological space (X, T), (X, ~o(T)) is FC(vii) iff there does not exist any non-constant continuous function firom (X, T) to I (with usual topology), a property which connectedness of (X, T) above cannot ~arantee. Our next obsezvation includes a character~mtion of FC(iv) (a~, hence also of FC(v), in View of Proposition 2.4 to follow next). ~~em 2.3. The following statements are equivalem for a fuzzy topological space (X, t):
(a) (x, t) is
cov)
(b) X canno~ be written as the supremum of two non-zero disjoint t-closed fuzzy sets.
(c) X cannot be written as the supremum of two non-zero disjoint t-open fuzzy sets,
(d) The only crisp t-clopen subset of X is 0 or 1. (e) There is no fuzzy continuous function f :(X, ~)--~l~ (or 1.~) other thun 0 and 1.
Proof. (a) :~ (b) :~ (c) :~ (d) is easy. (d) ~ (e) follows from the fact that if f :(X, t)-* G (or 1.~) is f u r y continuous with f ~0, f - 1, then f-1(1~) is proper crisp t-clopen, conU'adicting (d). Finally, to see ( e ) ~ (a), let (X, t) be not FC(iv). Then there exist two non-zero fuzzy sets ~s and y in X with X = ~ U y and f3 7 ffi0 = ~sN ~. Clearly, ~s and y must be crisp and t-clopen. Consider any function f :(X, t)-~/,4 (or I]) such thatf(~) ~_ IA and f(y) ~_ IA.. Then f-1(IA) IA of ----~ and f-t(IA.) = y showing that f is continuous. Evidently, f ~0, f ~ I, contradicting (e). In view of the fact that two crisp subsets are Q-separated iff they are separated, we get the following: l~s|deu
2.4. A fuzzy topological space (X, t) is FC(iv) iff it is FC(v).
We give another characterization of FC(iv) which is sin~lar to the one in Proposition 2.3 ((a) ~ (e)); its easy proof is omitted. P~pos|tton 2.$. A fuzzy topological space (X, t) is FC(iv) iff there is no non-constant fuzzy continuo~ funcffon f "(X, t)--~ 2v (or 2~).
Retook 2.3. The two characterizations of the fuzzy connectedness concept FC(iv) (--FC(v)), v~z. Proposition 2.3 ((a) ¢#(d)) and Proposition 2.5, are nearly identical with similar characterizations of eonnectedness in (usual) topology. All the fuzzy connectedne,,;sconcepts given in defii~ition2.1 are known to be
D. so. At/, A, K. S ~ ~ p r e s e ~ under ~ continuous functions. We now show that some of these ~ m ~ p r ~ r v ~ under some weber f o ~ s of fuzzy continuity alsoo 2.~ ~1]. Given a fuzzy topological space(X, t), a fuzzy set ~ in X is ~ ~ o p e n (rasp. t u g , closed) itt ~ = IntCl/~ (rasp. ~ = CI Int ~). A f ~ c ~ n f:(X, O-e.(y, s) between ~ topologicad spaces is called almost~zy co~nuou~ (resp~ ,.~,eaklyfuzzy c o ~ ~ ) ~ f - l ( l ~ ) e t for each regular open ~ n y ~ t ~ of (y, s) (~esp. ifff "~(/~)~_Intf-~(Cl/0for each/~ ¢ s).
2.6. The fuzzy con~ctedness concepts FC(i), FC(iv), and FC(vfi) are p~erved under both almost and weakly fuzzy continuousfunctions. Proof. We prove the asmrtion only for FC(iv), it being similar for FC(i) and ~(v~). Let f :(X, t)--~ (y, s) be an onto function between fuzzy topological spaces with (y, s)not being FC(iv). Then there exists a proper crisp s-clopen utbset, say/~, in y. Now ~ f is almost fuzzy continuous then as/~ is both regular open and regular c l o ~ , f-a(~) must be proper crisp t-clopen sub~t of X contradicting the fact that (X, t) is FC(~,v). Next, if f is weakly ~ continuous then f-x(/~)=_ Intf-t(Cl~)=IntJ-t(t~) so that f-a(#) is t-open. Similarly f - x ( ~ , ) g Intf-a(Cl ~') = Intf-l(/~ ') so that f-~'(/~') is also t-open. But f-l(/~,) ffi (f-l(~)),. Hence f-t(/,) is a proper crisp t-elopen subset of (X, t), a contradiction. We now give an example which shows that Proposition 2.6 above does not hold good for the remaining fuzzy eonnectedness concepts viz. FC(ii), FC(ifi) end FC(vi). Let Y denote the unit interval I with the fuzzy topology consisting of all constant fuzzy sets. Then Y is clearly FC(ii), FC(iii) and FC(vi). Let X denote the two-point set {a, b) with the fuzzy topology generated by all constant fuzzy sets, and ~ end y, where ~(a) ~ ~ ~ y(b) end/~(b) ---0 - y(a). The constant fuzzy set with value ½is ck~pen in X and (/~, 3') is (C1)- as well as (C2)-separated. So X is neither FC(ii) nor ~(iii). It is evidently not FC(vi) also. Now consider the function f: Y ~ X defined by a i f y E [ o , l], ¢(Y) = # if y E fl, 11.
We assert that y is both ~ o s t and weakly fuzzy continuous. That f is almost fuzzy continuous follows trivially firom the fact that the only regular open fuzzy ~:ets in X are constants. Next, since the ~zz,y ~.~osurei~ X of every open fuzzy set in X is a constant, it is easily verified that f is ~Iso weakly fuzzy continuous. The ~ eonneetedness concepts FC(i) and FC(~i), as noted earlier, are not good extensions of (topological) connectedness. It is shown in [3] that FC(ii) and lFC(i~) are good extensions of conneetedness. Our next result shows that the remarrying fuzzy connectedness concepts, viz, FC(i~) ( = FC(v)) and FC(vi) are ~so good extensions of connectedness.
On ~ z y conneaedness Pm~a or
2.7. A topologicM space (2(, T) is connected iff (X, co(T))~ FC~iv)
FC(,i).
~f. We prove the result only for FC(iv), the proof for FC(vi) being easier. First, let (X, T) be connected. If (X, co(T)) is not FC(iv) then there e~st two non-~ro w(T)-open ~ , ~/~~ y s u c h that Xffi/~U ~, ~ d ~ N y = 0 . Consequently, X ~ o U yo. Also, ~oN 1'o=0, Indeed, if x ~ Yo then ~(x) > 0 so we can choose r e I with y(x) > v > 0 whereby x e y-l(v, 1] G T. Now y-~(v, 1] cannot intersect /Jo, for if x'e~of3y-~(v, 1] then /~(x')>0 and y(x')>r, contradicting the fact that ~ N y = 0. Thus x cannot be a limit point of ~o i.e., x ¢ Po confirming that ~o N Yo= 0. Similarly ~o N ~o = 0. Thus (X, T) is disconnected, a contradiction. The implication (X, co(T)) is FC0v) =~ (X, T) is connected follows from the fact that if A ~ X, then CIe)(T)|A ~ l~r~. We close by presenting the various implications among the fuzzy connectedness concepts FC(i)-FC(vii) in the following proposition. The easy proof is omitted. It is not hard to construct counterexamples to show that no other implications ex~t among these concepts.
~ p o l t i o n 2.8. We have the following implications:
FC(v) FC(i) ~
FC(vii)
~ FC0v) ~
FC(fii)
FC(vO A¢_hew|edgement The authors would like to thank the referee for several pertinent commeat~.
Refe~ ~nces
[1] K.K. Azad, On ~xmy semieontinuity, fuzzy almost ¢ontimaity and fuzzy weakly continuity, J. Math. Anal. Pppl. 82 (1981) 14-32. [2] B. Hutton, Products ot fuzzy toxicologicalspaces, Topology Appl. 11 (1980) 59-67. [3] R. Lowen, Connectedness in ~ topological spaces, Rocky Mount~n J. Math. 11 (1981) 427-433. [4] R. Low©n, On the Existence of Natural Non.topological F ~ Topolo~cal Spaces (Helden~ann Ver|ag, ~t"l~, 1985). [5] P. M. Pu and Y. M. Liu, FuT~v topology l: Neighbourhood structure of a f ~ y pohlt e~nd Moore-Smith convergence, J. M~h. Anal. Appl. 76(1980) 571-599.
D. M. AU, A. K. Sr/vastava
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