two new l-fuzzy topologies on r (l)

b= sets and systems

ELSEVIER Fuzzy Sets and Systems 76 (1995) 379-385

Two new L-fuzzy topologies on R(L) Dong Yu

Department of Mathematics, Jinan University, Guangzhou, China

Received November 1993; revised March 1994

Abstract

In this paper, two new topologies 3'1, ~ on R(L) are constructed. Using ~i, we produced two L-fuzzy topologies r/i on R(L) other than the L-fuzzy topology T introduced by Hutton. We have proved that t/'l is finer than r/~ and t/~ is finer than T. r/2 is the coarsest among the induced L-fuzzy topologies which are finer than T. We have shown that (L RtL), ~h), the induced space of (R (L), z~), possesses many good properties, such as stronger separation, suitability, etc. We proved that closed interval ([a, b](L), r/2l[a, b](L)) is closed, connected and N-compact. The addition and multiplication defined on R(L) by Rodabaugh, are still jointly continuous.

Keywords: Fuzzy topology; Fuzzy real line; Connectedness; Induced space

1. Preliminaries

L-fuzzy unit interval I(L) plays an important role in L-fuzzy iopology theory. I(L) can be used to characterize L-fuzzy normality and uniformizability. It is the first standard, nongenerated L-fuzzy topological space. Because of technical difficulties, for a long time the understanding of I (L) and R (L) is meager. They attracted much attention of researchers. Lowen and Rodabaugh once concentrated on studying I(L) and R(L).

We made further analysis of I(L) and R(L), and found out that the L-fuzzy topology T introduced by Hutton is not as good as the Euclidean topology on [0, 1]. I(L) loses many good properties which [0, 1] has.

It is the purpose of this paper to construct two new L-fuzzy topologies t/~ and ~/~ on R(L), and show several good properties of (L R(L), t/'i). We also obtain the characterization of ~; in terms of nets, find out a copy of (R(L), ~2) in function space R~ x R~ and a copy of(R(L), zl ) in function space R ~ x R L and give the order of

I p rh, 1/2 and T. (L, v , ^ ,' ) denotes a completely distributive lattice, equipped with an order reversing involution ..... .0, 1

are the least and the greatest element of L, respectively. Furthermore, in this paper L is always a chain, L possesses the order topology.

Lemma 1. (Rodabough [-5]). Let ~ E L and [-2] ~ R ( L ) the followin~t hold: (1) There is a (2, ~)~ [ . - ~ , + c~] such that for any left continuous member of [2], 2(0 < ~ ' / f i t > a(2, c0.

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380 D. Yu / Fuzzy Sets and Systems 76 (1995) 379-385

(2) There is b (2, at)e [ - oo, + oo] such that for any right continuous member of[4] , 4(t) > m/fit < b(4, at). (3) The following are equivalent:

(i) to = a(4, 0t)[b(4, ~t)]; (ii) For each 4 e [ 4 ] , t > to [ < t o ] implies that 2(0 < ~' [>~t], and t < to [ > t o ] implies that

4(0/> ~' E~<~]; (iii) For some 4e [2], the conclusion of(ii) holds.

Let R = [ - ~ , + oo], Tt = { [ - o o , a) la e R}w{0}, T~ = {(a, + oo][a e R}u{0}, the topological space (R, Tt), (R, T~) is abbreviated to Re, R~, respectively. The topology generated by subbase Teu T~ is denoted by To. (R, To) is abbreviated to R.

Definition 1. Suppose {[2.]; neD} is a net in R(L), [4] eR(L), {[2.]; neD} W-converges to [2] if for any ~eL, {a(4., ~): neD} converges to a(2, ~) in Rt and {b(2.,~): neD} converges to b(A, ~t) in R~. We write l iminf[2.] = [2] in this case. {[2.]: neD} is called converged to [2], ifih R, for any ~eL, {a(2., ~t): neD} converges to a(2, ~t) and {b(2., at): neD} converges to b(A, ~t). We write lira I-An] = [2].

Let lira sup X., lim infX. be the largest and the least accumulation point of {X.: n e D}, respectively, then it is easy to verify that {a (2., n): n e D} converges to a (2, ~) in Re iff lira sup a (A., ~) ~< a (2, ~) and {b (2., 0t): n e D} converges to b(4, ~t) in R, iff b(2, ~) ~< lira infb(2., ~).

Definition 2. Let L be a chain, LI c L. Iffor any at, fieLd, ~t < fl, there exists 7 e L such that at < 7 < fl, then L1 is a dense subset of L. Put (at, fl) = {~lat < Y < fl}. If(~, fl) = 0, then (~t, fl) is a gap, L is called nongap if any (~,~) #0.

Definition 3. L-fts (L x, T) is connected if there does not exist U, V e T - {0, I}, such that U u V >0, Uc~V = O.

2. Properties of the convergences

Suppose r e R, let 2,: R --, L, 2,(0 = 1 when t < r else 4,(t) = 0, then [2,] e R (L).

Theorem 1. {r.: neD} is a net in R, reR , then l imr. = r / f f l im[2, . ] = [4,]; l imr. = r / ff l iminf[A,.] = 14,].

The proof is obvious and omitted. Theorem 1 shows that the convergences defined here are generalizations of convergence in real line R.

Theorem 2. Let L be a chain and possesses the order topology. I f lira inf[2 . ] = [2] (lim [2.] = [2]), then for 2.e [2.], Ae [2], {An(to): neD} converges to 2(to), where to is a continuous point erA(t).

Proof. Firstly, we prove that if l iminf[2.] = [2] then {An(to): neD} converges to 2(to), where to is a continuous point of 2(0. Let 2(to)e(0t, fl), then toe(a(2,ff), b(2, ~t)). Take rl , r2 eR such that a(2, f l ' )< rt < to < r~ < b(2, 0t). Since a(2, fl')/> l imsupa(2. , fl') and b(2, ct) ~< liminfb(2., at), there exists N e D such that for n > N, a(4., fl') < rl < to < r2 < b(4., ct), so 2.(to)e(at, fl). This implies that {4.(to): neD} converges to 4(to). []

The proof is similar in the case lim [2,] = [A-].

D. Yu / Fuzzy Sets and Systems 76 (1995) 379-385 381

Theorem 3. Let {[it.]: neD} and {[ke.]: neD} are nets in R(L) with the same domain, l f l iminf[ i t . ] = [it], lira inf[#.] = [/~] (lira [2.] = [it], lira [#.] = [/~]), then we have

(1) liminf[it.] ~ [#.] = [it] ~) [/~] (lira lit.] ~ [#,] = [2] ~ [#]); (2) Assume that there is a countable dense subset of L, then lira inf[it.]Q[/~.] = [it]Q[#]. (lira [it.](.~[/~.]

= [ i t ] o [~])

Proof. Analogue to that of Theorem 3 in [121. []

The following theorem is easy to verify.

Theorem 4. (1) For net {[2,]: neD}, where [2.] = [2]. We have liminf[it.] = [it]. (2) l f l iminf[ i t . ] = lit], then any subnet of {lit.]: neD} also w-converoes to [it]. (3) If{[it.]: n e D} does not w-converoes to lit], then there is a subnet of{lit.]: n e D }, its any subnet does not

w-converoe to [2]. (4) Let D be a directed set,for any m e D, Em is a directed set. Let F denote the set D x {Era: m e D} tooether

with the product order. For any ( m , f ) e F , put R ( m , f ) = (re, f (m)). I f for any reeD, neEm, S(m, n) is an element [it,.,.] e R(L) and lira inflim inf[it,.,.] = [2], then S o R also w-converoes to [2].

We have proved a similar result about the convergence defined by Definition 1. See [121.

3. Fopological space (R(L), ~a) and (R(L), z2)

Suppose A c R(L). Let C I ( A ) = {[it]lEit]eR(L), there exists a net in A converges to [it]}, C2(A)= {[it] I [it] e R(L), there exists a net in A w-converges to lit]}, then C1, C2 are closure operators by Theorem 4.

Definition 4. The collection of closed sets generated by Ci is denoted by zi, z l , z2 are cotopologies on R (L). Obviously, "[2 C= ~1"

Theorem 5. ~ :(R(L), z i )x(R(L) ,z i ) - - , ( (R(L) , zi) (i = 1,2) is continuous. Furthermore, assume there is a countable dense subset of L, then: Q :(R(L), zi) x (R(L), Ti) - , (R(L), zi) is continuous (i = 1, 2).

Proof. It follows from Theorem 3. [ ]

Lemma 2 (Rodabough [81). L e t a, b : L --, [ - oo, + oo1 such that the following conditions hold: (1) a is nondecreasino, b is nonincreasin9; (2) a(1)-- + o% b(1) = - ~ ; (3) a(ct), b(~t)e(-oo, + oo)for 0re {0, 1}; (4) a(0) <<, b(at) <<, a(~t') <<, b(O) for ~te{0, 1}; (5) b(=) = V {a(fl'): ~t < fl}, a(~t') = A{b(/~): fl < ~}.

If p: [ - 0% + oo] --+ L such that

f i t ~< a(O), p(t) -- {at: te[b(~), a(~t')l } a(O) < t ~< b(O),

t ~< b (0)


and we put 2 = p [ R, then 2 is a well-defined left continuous map, [2] e R (L) and for each ct e L, a (2, ~t) = a (~), b(2, ~) = b(~). []

According to Lemmas 1 and 2, there is a relation between R(L) and function space L L Re × Rr, put P = {(2 a, 2b)I [21 e (R (L), 2" (~) = a (2, ct), )b (a) = b (2, ct)}, then P c R~ x R~. If [21 ] -~ [),21, there exists ct e L, such that a(21, a) ~: a(22, ~). So the map G: [2] ~ ( 2 a, 2 b) is a bijeetion from R(L) to P. For K ~ R(L), put PK = {(2 ~, 2b) l [21 eK} c P, then PK is closed iff K is closed. Thus, (R(L), ~2) is homeomorphic to subspace (P, Z) of R~ × R~. (Z is the subspace topology).

I, e m m a 3 (Yu and Liu [121). Suppose [21 e R(L), then the following hold: (1) [2] e [a, b ] (L) / f f a(2, 0), b(2, 0)e [a, bl, (2) [2] e (a, b)(L)/ff [2] e [a, b] (L) and for any a e L - {0, 1 }, a (2, a), b (2, a) e (a, b); (3) [2] e(a, b] (L) /ff [2] e [a, hi(L) and for any a e L - {1}, b(2, a)e(a, hi; (4) [2] e [a, b) (L) / f f [2] e [a, b] (L) and for any ct e L - {0}, a(2, a) e [a, b).

Lemma 4 (Yu [13]). Suppose al (~), bl (a) and a2 (a), b2 (~) are pairs of functions satisfyin O the conditions of Lemma 2. For s, t e R, s > O, t > O, a(o 0 = sal (a) + ta2(a), b(a) = sbl (a) + tb2(a) also satisfy the conditions of Lemma 2.

Theorem 6. (R (L), z2) and its subspaces (a, b)(L), [a, b] (L), (a, b] (L), [a, b)(L) are path connected To spaces. [a, b] (L) is closed.

Proof. Since Re and Rf are To spaces, the subspace P ofR~ x R~ are To. (R(L), ~2) ~ (P, Z). So (R(L), ~2) and its subspaces are To.

For any [2 ]e C2([a, b](L)), there exists a net {[2.]: neD} in [a, b](L) which w-converges to [2]. So a(2., 0), b(2,, 0)e [a, b] by Lemma 3, then the following hold:

a ~< lira supa(2n, 0) ~< a(2, 0) ~< b(2, 0) ~< l iminfb(2. , 0) ~< b.

Thus [2] e [a, b] (L) implying [a, b] (L) is closed. Obviously, [a, a] (L) is path connected. Assume a < b, then [a, b](L) - [0, 1](L), (a, b)(L) ~ (0, 1)(L), (a, b](L) _-__ (0, 1](L), [a, b)(L) ~- [0, 1)(L), R(L) - (0, 1)(L). We only need to show that (0, I)(L), [0, 1)(L), (0, 1](L) and [0, 1](L) are path connected.

For [21], [22]e[0 , 1](L), se [0 , 1], by Lemmas 2 and 4, there is a [ps]eR(L) such that for any a a L , a(ps, a) = sa(21, ~) + (1 - s)a(2z, a), b(ps, a) = sb(21, at) + (1 - s ) b ( 2 2 , a). S i n c e a(~.i, 0), b(~. i, 0 ) e [0 , 1], a(p~,O), b(p~,0)e[0,1] . Put O(S)= [p~], 9 : [0 ,1]- -*[0 , 1](L), then 9 is well defined and 9(0)= [22], 9(1) = [21]. I fs . , se [0 , 11 (n = 1, 2 . . . . ), s , - -*s (n~ oo), then:

lira sup a(p~, ct) ~< lim sup s.a(21, ct) + lim sup(1 - sn)a(22, ~t) = sa(2~, ~t) + (1 - s)a(,,].2, ~),

liminfb(p~, ~t) 1> l iminfs.a(21, ~t) + liminf(1 - s,)a(22, ~) = sa(21, ~) + (1 - s)a(22, ct).

So lira g(s.) = O(s). It follows that 9 is continuous, thus [0, 1] (L) is path connected. For [21], [22] e(0, I)(L), se [0, 1], construct [p~] as above. Noticing that

rain {a(2~, ~)} ~< sa(2,, ct) + (1 - s)a(22, ~) ~< max {a(2, ~)},

rain {b(2, ct)} ~< sb(2~, ct) + (1 - s)b(22, ~) ~< max{b(2~, a)}.

For any cteL - {0, 1}, a(2i, ~), b(2i, ct)e(0, 1), so a(p~, ~), b(p~, ~t) e(0, 1). Hence [p~] e(0, 1)(L) by Lemma 3. Let O(s)= [ps], g:[0, 1] ~ (0 , I)(L), then 9 is continuous, 9 (0)= [22], 9 (1)= [21]. So (0, 1)(L) is path connected. Similarly, (0, 1](L) and [0, 1)(L) are path connected. []


Theorem 7. (R (L), z~) and its subspaces (a, b)(L), [a, b] (L), (a, b] (L), [a, b)(L) are T7/2 and path connected spaces. [a, b] (L) is closed.

Though r~ is finer then z2, Zl is not too fine to break the path connectedness.

4. L-fuzzy topological spaces (L s ~L~, ~ 1 ) and (L R ~L~, t/2)

In [3], the authors skillfully proved that completely distributive law is sufficient to construct the topological structure of the lattice value semicontinuous mappings. For the sake of expression, we use cotopology.

In the following, t/denotes the collection of closed subsets of L-fts (L x, ~l). The family of all crisp closed sets in r/is denoted by [~/].

Definition 5. L-fts (L x, tl) is called fully stratified if each constant in L belongs to t/, (L x, tl) is called weakly induced, if for each F ~ ~/, any ~ ~ L, F, = {x I x ~ X, F (x) >>. ct} ~ rl. (L x, 7) is called induced if (L x, tl) is both fully stratified and weakly induced.

Now, consider the induced space of (R (L), ~1). Let th = {F I F: R (L) --* L, ct ~ L, F, e zi }, then (L ~lz), r/l) and (L Ra), t/2) are induced spaces. To study the relation between T and qi, we need the following lemma.

Lemma 5. Let L be a chain, [2] ~R(L), then the following hold: (1) t < b(2, ~) / f f2( t+) > ~. (2) s > a(2, ct)/ff2(s_) < a.

Proof. (1) If t < b(2, ct), take tl e(t, b(2, ~)), then 2(t+)/> 2(tl) > ~. If 2(t+) > a, choose right continuous 2~ [2], then 2(0 = 2(t+) > a. So t < b(2, ~t).

(2) Analogue to (1). []

Theorem 8. tl'l D if2 ~ T (generated by {Ls, R, Is, t E R} ). I f (L R{L), U) is induced, U' = T, then U ~ th. This means if2 is the coarsest L-fuzzy topology among the L-fuzzy topologies which are induced and finer than T.

Proof. For R,, Ls ~ T, R't and L'~ are T-closed, for any a e L

(R~), = {,~ I [,~3 ER(L), 2(t+) ~< ~'} = {[~31 [,~3 ~R(L), b(2, a') % t} ~z,,

(Lj), = {~ I [,~3 ~ R(L), ).(s_ ) i> a} = {E,~] I E23 ~R(L), a(2, a) t> s} ~ z,.

Hence T ~ t/[ ~ r/~. If (L R~L), U) is an induced space and U' ~ T, then we have

(R~)~ = {21 [).] ~R(L ), b(2, a') ~< t } ~ [ U ] , a c E ,

(Lj)~ -- {21 [2] eR(L) , a(2, ~) >1 s } e [ U ] , a~L.

There exists a bijection between R (L) and P c R L x R L. So the cotopology [U ] on R (L) can be transfered to P. We denote this cotopology by Uo. Hence (R(L), [U] ) -~ (P, Uo). We know that

{(f, g)I B ~ L , g(a)~ [ - oo, t ]}nP~uo,

{(f, a) l B ~ L , f ( a ) e [s, + oo]}c~P~uo,

but these sets are members of a subbase of the cotopology Z, so Z c Uo. Since (R(L), z2) ~ (P, Z) and z2 ~ [U] , we have t/2 c U.


Theorem 9. (L atL), ~li) is suitable, A c R(L), tAI >1 2, then subspace (L A, ~l~lA) is suitable (i = l, 2). []

Theorem 10. t~ :(L RtLJ, Fh) x (L RtL~, tli) ~ (L RtL~, rh) is L-fuzzy continuous. I f L has a countable dense subset, then Q :(L RtLJ, ~/i) x (L RtL~, ~h) -* (LRtL), ~h) is L-fuzzy continuous, (i = 1, 2).

Proof. Immediate result from Theorem 5 and Proposition 5 of [3]. []

In [3], to study compactification, the authors introduced a separation axiom scheme. We have following results on this scheme. []

Theorem l l . (L stL), ~h ) is a T*/2 space.

Proof. First of all, we prove (L RtL~, ~/1) is T1. For any singleton/'~

{ ~, at > r,

(P~)~ = x, O < ~ r ,

R(L), a = O.

So /Y~ is closed, implying (LRtL~,th) is TI, we conclude that (LRtLJ, th) is T¢/2 from Theorem 7 and Proposition 8 of [3]. []

When L is nongap, we generalize some results of [9] and use these results to solve some problems concerning (L Ra°, t/i) and their subspaces with respect to each of the separation axiom schemes of Hutton and Sarkar. It is easy to verify the following lemma.

Lemma 6. I f L is a nongap chain, A ~ L, then we have

A = (Jr Ar = ~ r Ar

where Ar = { x l x ~ X , A(x) >1 r}, A.r = { x l x ~ X , A(x) > r}.

Theorem 12. lnduced L-fts (L x, tl) is Ti iff (X, [r/]) is Ti if L is nonoap (i = 1, 2, 3).

Proof. Analogous to those of [9]. []

Theorem 13. I f L is nongap, then (L Ra°, rh ) and its subspaces are Ta,(L R~L~, t/2) and its subspaces are To.

About separation axiom of Sarkar, we have the following result.

Theorem 14. (L Ra~, ~h ) and its subspaces are T3 in sense of Sarkar.

Proof. It follows from Theorems 13 and 5.1 in [7]. []

Now, we study the connectedness of (L RtL), th) and its subspaces.

Theorem 15. Induced space (L x, ~l) is connected iff (X, [t/]) is connected.

The proof is trivial.


Theorem 16. (L RtL~, rh) and subintervals [a, b] (L), (a, b)(L), [a, b)(L), (a, b] (L) are connected. (O-connected, Q- connected) (i = 1, 2).

Corollary. (L ~tL), T) and its subintervals are connected.

Theorem 17. f : (L ~L), r/l) ~ (L RtL), th) ( f :(L RtLI, r/2) ~ (L RtL), r/2)) is L-fuzzy continuous ifffor any {[An]: n ~ D}, lira [2n] = [2]. (lim inf[2~] = [2]) always implies lim f([2n]) =f ( [2] ) , (lira inf f([2n]) =f([2])) .

Proof. According to Proposition 5 of [3], f : ( L mL~, rh) ~ (L mL~, ~li) is L-fuzzy continuous i f f f : (R(L), zi) (R(L), zi) is continuous. So the conclusion is followed. []

Remark. This paper summarize the author's recent research work on L-fuzzy real line R(L). Some scholars achieved similar results by other approach. In [101, Wang Guo-Jun and Xu Luo-shan introduce a new L-fuzzy topology on I(L). Let C denote the family of constant mappings. It turns out that r/: = ~wC, they remove the restriction that L is a chain, but they do not obtain the characterization of 2 (3.) in terms of nets. Since L-fts (L x, z) is N-compact iff (L x, TwC) is N-compact and (L x~LI, 3 ) is N-compact [10], (L R~L~, r/2) is N-compact.

Reference

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(1985) 285-310. [9] G. Wang and L. Hu, On induced fuzzy topological spaces, J. Math. Anal. Appl. 108 (1985) 495-506.

[10] G.-j. Wang and L.-s. Xu, Inner topology and refinement of Hutton unit interval I(L), Sci. Sinica 7 (1992) 705-712. 1-11] D. Yu and W. Liu, Sequence convergence in L-fuzzy real line R(L), d. Sichuan Normal Univ. 4 (1987) 40-44 [12] D. Yu and W. Liu, An ideal L-fuzzy topology on R(L), Fuzzy Systems and Math. 12 (1988) 30-35. [13] D. Yu, The connectedness of R(L) and I(L), Sichuan Normal Univ. 1 (1989) 87-90. [14] D. Yu, Generalized order homomorphism and continuity lemma, J. dinan Univ. (3) (1989), 15-21. [15] C. Zheng, Fuzzy path and fuzzy connectedness, Fuzzy Sets and Systems 14 (1984) 273-280.

two new l-fuzzy topologies on r (l)

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