On R1 Space in L-Topological Spaces

GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 37 (2017) 63-71

ON R1 SPACE IN L-TOPOLOGICAL SPACES

Rafiqul Islam1,* and M. S. Hossain2 1Department of mathematics, Pabna University of Science and Technology, Pabna-6600, Bangladesh. 2Department of Mathematics, University of Rajshahi, Rajshahi, Bangladesh. *Corresponding author: [email protected]

Received 27.12.2016 Accepted 30.07.2017

ABSTRACT

In this paper, R1 space in L-topological spaces are defined and studied. We give seven definitions

of R1 space in L-topological spaces and discuss certain relationship among them. We show that all of these satisfy ‘good extension’ property. Moreover, some of their other properties are obtained.

Keywords: L-fuzzy set, L-topology, Hereditary, projective and productive.

1. Introduction

The concept of R1-property first defined by Yang [19] and there after Murdeshwar and Naimpally [15], Dorsett [6], Dude [7], Srivastava [17], Petricevic [16] and Candil [11]. Chaldas et al [4] and

Ekici [8] defined and studied many characterizations of R1-properties. Later, this concept was generalized to ‘fuzzy R1-propertise’ by Ali and Azam [2, 3] and many other fuzzy topologists. In this paper we defined seven notions of R1 space in L-topological spaces and we also showed that this space possesses many nice properties which are hereditary productive and projective.

2. Preliminaries

In this section, we recall some basic definitions and known results in L-fuzzy sets and L-fuzzy topology.

Definition 2.1. [20] Let be a non-empty set and = [0, 1]. A fuzzy set in is a function : → which assigns to each element ∈ , a degree of membership, () ∈ .

Definition 2.2. [9] Let be a non-empty set and be a complete distributive lattice with 0 and 1. An L-fuzzy set in X is a function : → which assigns to each element ∈, a degree of membership, () ∈ .

Definition 2.3. [14] An L-fuzzy point in is a special L-fuzzy sets with membership function

() = =

() =0 ≠ where ∈. Definition 2.4. [14] An L-fuzzy point is said to belong to an L-fuzzy set in ( ∈ ) if and only if () < () and () ≤(). That is ∈ <(). 64 Islam and Hossain

Definition 2.5. [10] Let be a non-empty set and L be a complete distributive lattice with 0 and 1. Suppose that be the sub collection of all mappings from to .. ⊆ .Then is called L- topology on if it satisfies the following conditions:

(i) 0∗ , 1∗ ∈

(ii) If , ∈ then ∩ ∈

(iii) If ∈ for each ∈∆ then ∪∈∆ ∈. Then the pair (, ) is called an L-topological space (lts, for short) and the members of are called open L-fuzzy sets. An L-fuzzy sets is called a closed L-fuzzy set if 1−∈.

Definition 2.6. [20] An L-fuzzy singleton in is an L-fuzzy set in which is zero everywhere except at one point say , where it takes a value say with 0<≤1 and ∈. The authors denote it by and ∈ iff ≤().

Definition 2.7. [14] An L-fuzzy singleton is said to be quasi-coincident (q-coincident, in short) with an L-fuzzy set in , denoted by iff +() >1 . Similarly, an L-fuzzy set in is said to be q-coincident with an L-fuzzy set in , denoted by if and only if () +() >1 for some ∈ . Therefore iff () +() ≤1 for all ∈ , where denote an L- fuzzy set in is said to be not q-coincident with an L-fuzzy set in .

Definition 2.8. [3] Let : → be a function and be an L-fuzzy set in . Then the image () is an L-fuzzy set in whose membership function is defined by

(())() =sup ()() = if () ≠ ∅, ∈

(())() = 0 if () = ∅, ∈ .

Definition 2.9. [2] Let be a real-valued function on an L-topological space. If {: () >} is open for every real , then is called lower-semi continuous function (lsc, in short).

Definition 2.10. [14] Let (, ) and (, ) be two L-topological space and be a mapping from (, ) into (, ) . . : (, ) → (, ). Then is called

(i) Continuous iff for each open L-fuzzy set ∈⟹() ∈ . (ii) Open iff () ∈ for each open L-fuzzy set ∈. (iii) Closed iff () is s-closed for each ∈ where is closed L-fuzzy set in . (iv) Homeomorphism iff is bijective and both and are continuous..

Definition 2.11. [14] Let be a nonempty set and be a topology on . Let =() be the set of all lower semi continuous (lsc) functions from ( , ) to (with usual topology). Thus () = { ∈ :(, 1] ∈ } for each ∈. It can be shown that () is a L-topology on . Let “P” be the property of a topological space ( , ) and LP be its L-topological analogue. Then LP is called a “good extension” of P “if the statement ( , ) has P iff (, ()) has LP” holds good for every topological space (, ). On R1 Space in L-Topological Spaces 65

Definition 2.12. [18] Let (,) be a family of L-topological spaces. Then the space (Π,Π) is called the product L-topological space of the family of L-topological space {(,): ∈∆} where

Π denote the usual product of L-topologies of the families {: ∈∆} of L-topologies on . An L-topological property ‘P’ is called productive if the product L-topological space of a family of L-topological space, each having property ‘P’ also has property ‘P’.

A property ‘P’ in an L-topological space is called projective if for a family of L-topological space

{(,): ∈∆}, the product L-topological space (Π,Π) has property ‘P’ implies that each coordinate space has property ‘P’.

Definition 2.13. [1] Let (, ) be an L-topological space and ⊆. we define ={|:∈} the subspace L-topologies on induced by . Then (, ) is called the subspace of (, ) with the underlying set .

An L-topological property ‘P’ is called hereditary if each subspace of an L-topological space with property ‘P’ also has property ‘P’.

3. R1-property in L-Topological Spaces

We now give the following definitions of R1-property in L-topological spaces. Definition 3.1. An lts ( , ) is called

(a) −() if ∀ , ∈ , ≠ whenever ∃ ∈ with () ≠ ()then ∃ , ∈ such that () =1,() =0,() =0,() =1 and ∩=0.

(b) −() if ∀ ,∈,≠ whenever ∃ ∈ with () ≠ () then for any pair of

distinct L-fuzzy points , ∈() and ∃ , ∈ such that ∈, ∉ and ∉, ∈ , ∩=0.

distinct L-fuzzy singletons , ∈(), and ∃ , ∈ such that ⊆, and

⊆, and ∩=0.

(d) −() if ∀ , ∈ , ≠ whenever ∃ ∈ with () ≠ () then for any pair of

distinct L-fuzzy points , ∈() and ∃ , ∈ such that ∈, and ∈ , and ∩=0.

(e) −() if ∀ , ∈ , ≠ whenever ∃ ∈ with () ≠ () and for any pair of

distinct L-fuzzy points , ∈() and ∃ , ∈ such that ∈⊆, ∈⊆ and ⊆.

(f) −() if ∀ ,∈,≠ whenever ∃ ∈ with () ≠ () then ∃ , ∈ such that () >0,() =0 and () =0,() >0 .

(g) −() if ∀ , ∈ , ≠ whenever ∃ ∈ with () ≠ () then ∃ , ∈ such that () >() and () >(). 66 Islam and Hossain

Here, we established a complete comparison of the definitions

−(),−(),−(),−(),−() and − () with −(). Theorem 3.2. Let ( , ) be an lts. Then we have the following implications:

− () − ()

− ()  − () − ()

− () −()

The reverse implications are not true in general except −() and −().

Proof: −() ⇒−() , − () ⇒−()can be proved easily. Now −() ⇒

−() and −() ⇒−(), since −() ⇔−()and −() ⇔−

(). −() ⇒−(); It is obvious. −() ⇒−(), since −() ⇒−

().

The reverse implications are not true in general except −() and −(), it can be seen through the following counter examples:

Example-1: Let ={,}, be the L-topology on generated by {: ∈ } ∪ {, , } where () =0.6,() =0.7,() =0.5,() =0,() =0,() =0.6

= {0,0.05,0.1,0.15, … … … 0.95,1} and = 0.4, = 0.3.

Example-2: Let ={,}, be the L-topology on generated by {: ∈ } ∪{,,} where () =0.8,() =0.9,() =0.5,() =0,() =0,() =0.4

= {0,0.05,0.1,0.15, … … … 0.95,1} and = 0.5, = 0.4.

Proof: − () ⇏−(): From example-1, we see that the lts (, ) is clearly −() but it is not −(). Since there is no L-fuzzy set in which grade of membership is 1.

−() ⇏−(): From example-2, we see the lts (, ) is clearly −() but it is not

−(). Since −() ⇏−() and −() ⇏−() so −() ⇏−

().

−() ⇏−(): This follows automatically from the fact that

−() ⇔−() and it has already been shown that −() ⇏

−() so −() ⇏−().

−() ⇏−(): Since −() ⇔−()and −() ⇏−() so −

() ⇏−(). But −() ⇒−() ⇒

−() is obvious.

On R1 Space in L-Topological Spaces 67

4. Good extension, Hereditary, Productive and Projective Properties in L-Topology

We show that all definitions −(), −(),−(),

−(),−(),−() and −() are ‘good extensions’ of −property, as shown below:

Theorem 4.1. Let ( ,) be a topological space. Then ( , ) is iff ( , ()) is −(), where = , , , , , , .

Proof: Let ( , ) be . Choose , ∈ , ≠. Whenever ∃ ∈ with ∈ , ∉ or ∉ , ∈ then ∃ , ∈ such that ∈,∉ and ∈,∉ and ∩=∅. Suppose

∈,∉ since ∈ then 1 ∈() with 1() ≠ 1(). Also consider the lower semi continuous function 1,1, then 1,1 ∈() such that 1() =1,1() =0 and 1() =

0, 1() =1 and so that 1 ∩1 =0 as ∩=∅. Thus ( , ()) is −().

Conversely, let ( , ()) be −(). To show that ( , ) is . Choose , ∈ with ≠. Whenever ∃ ∈ with () ≠() then ∃ , ∈ () such that () =1,() =0,() = 0, () =1 and ∩=0. Since () ≠(), then either () <() or () >(). Choose () <(), then ∃ ∈ such that () <<(). So it is clear that (, 1] ∈ and ∉(, 1], ∈(, 1]. Let ={1} = {1}, then , ∈ and is

∈,∉ ,∉,∈, and ∩=∅ as ∩=0 . Hence ( , ) is .

Similarly, we can show that −(),−(),−(),

−(),−(),−() are also hold ‘good extension’ property.

Theorem 4.2. Let ( , ) be an lts, ⊆ and ={ǀ:∈}, then

(a) ( , ) is −() ⇒(,) is −().

(b) ( , ) is −() ⇒(,) is −().

(d) ( , ) is −() ⇒(,) is −().

(e) ( , ) is −() ⇒(,) is −().

(f) ( , ) is −() ⇒(,) is −().

(g) ( , ) is −() ⇒(,) is −().

Proof: We prove only (a). Suppose ( , ) is L-topological space and is also −().We shall prove that (, ) is −(). Let , ∈ with ≠ and ∃ ∈ such that () ≠(), then , ∈ with ≠ as ⊆. Consider be the extension function of on X, then

() ≠(), Since ( , ) is −(), ∃ , ∈ such that () =1,() =0,() =

0, () =1 and ∩=0. For ⊆, we find ,ǀ, ǀ ∈ and ǀ() =1,ǀ() =0 and ǀ() =0,ǀ() =1 and ǀ ∩ ǀ = (∩)ǀ = 0 as , ∈ . Hence it is clear that the subspace (, ) is −(). Similarly, (b), (c), (d), (e), (f), (g) can be proved. 68 Islam and Hossain

So it is clear that −(),=,,…, satisfy hereditary property.

Theorem 4.3. Given {(,): ∈Λ} be a family of L-topological space. Then the product of L- topological space (Π,Π ) is −() iff each coordinate space (,) is −(), where = , , , , , , .

Proof: Let each coordinate space {(,): ∈Λ} be −(). Then we show that the product space is −(). Suppose , ∈ with ≠ and ∈Π with () ≠ (), again suppose

=Π,=Π then ≠ for some ∈Λ.But we have () =min {(): ∈ Λ}, and () =min {(): ∈ Λ}. Hence we can find at least one ∈ with ≠, since each (,): ∈Λ be −() then ∃ , ∈ such that = 1, =0,= 0, =1 and ∩ =0. Now take =Π′, =Π′ where = , = and =

=1 for ≠. Then , ∈ Π such that () = 1, () = 0,() =0,() =1and ∩= 0. Hence the product of

L-topological space is also L-topological space and (Π,Π ) is −().

Conversely, let the product L-topological space (Π,Π ) is −(). Take any coordinate space ,, choose , ∈ , ≠ and ∈Π with () ≠(). Now construct , ∈ such that =Π′, =Π′ where = for ≠ and =, =. Then ≠ and using the product space −(), Π ∈Π with Π() ≠Π(), since (Π,Π ) is

−() then ∃ , ∈ Π such that () =1,() =0,() =0,() =1 and ∩=0. Now choose any L-fuzzy point in . Then ∃ a basic open L-fuzzy set Π ∈Π such that ∈Π ⊆ which implies that <Π () or that <

and hence <Π ∀ ∈Λ……() and

() =0⇒ Π() =0 ……().

Similarly, corresponding to a fuzzy point ∈ there exists a basic fuzzy open set Π ∈Π such that ∈Π ⊆ which implies that

< ()∀ ∈Λ……() and

Π () =0……(). Further, Π () =0⇒ () =0, since for ≠, = and hence from (), = >. Similarly, Π () =0⇒ () =0 using ().

Thus we have () >, () =0 and () >, () =0. Now consider = , =, then () =1,() =0,() =0,() =1 and ∩ =0, showing that (,) is −(). Moreover one can easily verify that

(,), ∈Λ is −() ⇔ (Π,Π ) is −().

(,), ∈Λ is −() ⇔ (Π,Π ) is −(). On R1 Space in L-Topological Spaces 69

(,), ∈Λ is −() ⇔ (Π,Π ) is −().

Hence, we see that −(),−(),−(),−(),

−(),−(),−() Properties are productive and projective.

5. Mapping in L-topological spaces

We show that −() property is preserved under one-one, onto and continuous mapping for = , , , , , , .

Theorem 5.1 Let (, ) and (, ) be two L-topological space and : (, ) → (, ) be one-one, onto L-continuous and L-open map, then

(a) ( , ) is −() ⇒(,) is −().

(b) ( , ) is −() ⇒(,) is −().

(d) ( , ) is −() ⇒(,) is −().

(e) ( , ) is −() ⇒(,) is −().

(f) ( , ) is −() ⇒(,) is −().

(g) ( , ) is −() ⇒(,) is −().

Proof: Suppose (, ) is −().We shall prove that (, ) is −(). Let , ∈ with

≠ and ∈ with ()≠(). Since is onto then ∃ , ∈ such that () = and () = , also ≠, as is one-one. Now we have () ∈ , Since is L- continuous, also we have ()() =() =() and ()() =() = ().Therefore ()() ≠ ()(). Again since (, ) is −() and ∃ () ∈ with ()()≠ ()() then ∃ , ∈ such that () =1,() =0, () =0,() =1 and ∩=0. Now

()() ={():()=}=1

()() ={():()=}=0

()() ={():()=}=1 And

(∩)() ={sup(∩)():()=

(∩)() ={sup(∩)():()= 70 Islam and Hossain

Hence (∩) =0⇒() ∩() =0

Since is L-open, (),()∈. Now it is clear that ∃ (),()∈ such that ()() =1

, ()() =0,()() =0, ()() =1 and () ∩() =0. Hence it is clear that the

L-topological space (, ) is −(). Similarly (b), (c), (d), (e), (f), (g) can be proved.

Theorem 5.2 Let (, ) and (, ) be two L-topological spaces and :(,)→(,) be L- continuous and one-one map, then

(a) ( , ) is −() ⇒(,) is −().

(b) ( , ) is −() ⇒(,) is −().

(d) ( , ) is −() ⇒(,) is −().

(e) ( , ) is −() ⇒(,) is −().

(f) ( , ) is −() ⇒(,) is −().

(g) ( , ) is −() ⇒(,) is −().

Proof: Suppose (, ) is −().We shall prove that (, ) is −(). Let , ∈ with

≠ and ∈ with ()≠(), ⇒()≠() as is one-one, also () ∈ as is L- open. We have ()(()) = sup {()} and ()()=sup {()} and ()(()) ≠

()(). Since (, ) is −(),∃ , ∈ such that () = 1, (()) = 0, ()=0,(()) = 1 and ∩=0. This implies that ()() =1, ()()= 0, ()() =0, ()()=1 and (∩) =0⇒ () ∩ () =0.

Now it is clear that ∃ (), () ∈ such that ()() =1, ()()=0, ()() =0, ()() =1 and () ∩ () =0. Hence the L-topological space

(, ) is −(). Similarly (b), (c), (d), (e), (f), (g) can be proved.

REFERENCES [1] Abu Safiya, A. S., Fora , A. A. and Warner, M. W., Fuzzy separation axioms and fuzzy continuity in fuzzy bitopological spaces. Fuzzy sets and systems, 62 (1994), 367-373.

[2] Ali, D. M. and Azam, F. A., On Some R1-Properties in Fuzzy Topological Spaces. Journal of Scientific Research, 4 (1) (2012), 21-32.

[3] Ali, D. M. and Azam, F. A., Some Remarks on Fuzzy R0, R1 and Regular Topological Spaces. Journal of Scientific Research, 4 (2) (2012), 327-336.

[4] Caldas, M., Jafari, S. and Noiri, T., Characterization of pre R0 and R1 topological spaces. Topology proceeding, 25(2000), 17-30. [5] Chang, C.L., Fuzzy topological spaces. J. Math. Anal Appl., 24(1968), 182-192.

[6] Dorsett, C., R0 and R1 topological spaces. Math. Vesnik, 2(15) (30)(1978), 112-117.

[7] Dude, K.K., A note on R1-topological spaces. Period. Math. Hungar, 13(4) (1982), 267-271. [8] Ekici, E., On R spaces. int. j. pure. Appl. Math., 25(2) (2005), 163-172. On R1 Space in L-Topological Spaces 71

[9] Goguen, J.A., L-fuzzy sets. J. Math. Anal. Appl., 18(1967), 145-174. [10] Jin-xuan, F. and Ren Bai-lin, A set of new separation axioms in L-fuzzy topological spaces. Fuzzy sets and Systems, 96 (1998), 359-366.

[11] Kandil, A. and Sedahmed, E.H., A fuzzy R1 separation axioms. Information Sciences, 79(1994), 293- 313.

[12] Keskin, A. and Nori, T., On − and − spaces. Miskole Mathematics Notes, 10(2) (2009), 137-143. [13] Li, S.G., Remarks on product of L-fuzzy topological spaces. Fuzzy Sets and Systems, 94(1998), 121- 124. [14] Liu Ying-Ming and Luo Mao-Kang, Fuzzy Topology. Copyright © by World Scientific Publishing Co., Pte. Ltd. 1997.

[15] Murdeshwar, A.G. and Naimpally, S.A., R1-topological spaces. Canad. Math. Bull., 9(1965), 521-423.

[16] Petricevic, Z., R0 and R1 axioms in fuzzy topology. Mat. Vesnik, 41(1989), 21-28.

[17] Srivastava, A.K., R1-fuzzy topological spaces. J. Math. Anal Appl., 127(1987), 151-154.

[18] Srivastava, M. and Srivastava, R., On fuzzy pairwise – T0 and fuzzy pairwise – T1 Bitopological spaces. Indian J. Pure appl. Math., 32(3)(2001), 387-396. [19] Yang, C. T., On Para compact spaces. Proc. Amer. Math. Soc., 5(1954), 185-189. [20] Zadeh, L.A., Fuzzy sets. Information and Control, 8 (1965), 338-353.