IJRIT International Journal of Research in Information Technology, Volume 2, Issue 6, June 2014, Pg: 7-14
International Journal of Research in Information Technology (IJRIT) www.ijrit.com
ISSN 2001-5569
Fuzzy Grill m-Space and Induced Fuzzy Topology 1
Rashmi Singh1, Sonal Mittal2 Assistant Professor, Amity Institute of Applied Sciences, Amity University Noida, U. P, India
[email protected] 2
Research Scholar, Amity Institute of Applied Sciences, Amity University Noida, U. P, India
[email protected]
Abstract. The purpose of this paper is to introduce the notions of two operators and induced by fuzzy grill G on a fuzzy m- space. A new is obtained with the help of non- fuzzy topological space (fuzzy m-space) and the basic properties of the induced fuzzy topology topology are studied. Then a sort of suitability condition of fuzzy grill with fuzzy m- space is formulated. Keywords: Fuzzy grill, - operator, – operator
1. Introduction The concept of Grill was introduced by Choquet [3] in 1947 since then it has been extensively used by many authors, grill has subsequently turned out to be a very convenient tool for various topological investigations. It is also seen from the literature that in many situations, grill are more effective than certain similar concepts like nets and filters .Chang [2] introduced the notion of fuzzy topology in 1968. In fuzzy setting, the concept of fuzzy grills on fuzzy topological spaces was initiated by Azad [4]. Subsequently Srivastava and Gupta [10] investigated fuzzy basic proximity by use of fuzzy grill. The concept of complete grill in an L- merotopy, a concept similar to that of a bunch in the classical theory and an Lmerotopy has been introduced by Mona Khare and Rashmi Singh [5,6,7]. In 2010, M. Khare and S. Tiwari studied grill determined L- approach merotopological spaces In 2005, Bhattacharya, M N Mukherjee and S P Sinha have studied fuzzy compactness fuzzy almost compactness via fuzzy grill. Roy and Mukherjee [1] introduced an operator defined by grill on topological spaces. They [9] introduced the operators by fuzzy grill on fuzzy topological showed that it satisfy Kuratowski‘s closure axioms. In 2010, Sunita Das and M .N. Mukherjee introduced a kind of generalized fuzzy closed sets in terms of fuzzy grill and discussed their basic properties. In 2013[11] S. Modak introduced operators on grill m space. In the present paper, we have introduced fuzzy m-space and fuzzy grill on fuzzy m-space. We obtained a fuzzy-topology, however fuzzy m-space need not be a fuzzy topological space. Here we have studied certain basic properties of this new induced fuzzy topology. In section 3, we introduce two fuzzy operators and . In section 4, we define a suitability condition which when imposed on fuzzy grill G with fuzzy m-space makes generated fuzzy topology better behaved and applicable. Finally, we undertake a few fuzzy topological concepts for their study in respect of the m –space vis-à-vis the induced topology. Throughout this paper, we used the following definitions.
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2. Preliminaries
And
A fuzzy set A in a set X is a function on X in to the closed unit interval of the real line. Support of is denoted by Supp A. The fuzzy sets in X taking on respectively the constant values 0 and 1 are denoted by and respectively. For if for each . notation means that A is quasi-coincident with B i.e. implies for some .The negations of these statements are denoted by .A fuzzy singleton or a fuzzy point with support X and value denoted by .A is called of B of for some fuzzy open set U in X with , if A itself is fuzzy open then it is called an openof B. The collection of all open of any fuzzy point is denoted by ( ).A subfamily of the fuzzy topology of an fts is a base for iff for each fuzzy singleton in and for each open U of , for some with is called an adherence point of fuzzy set A if every open of is a quasi – coincident with A and is the union of all adherence points of A. is set of molecules of in A. , for where , P is a Poset.
3. Fuzzy grill on Fuzzy m-space Definition 3.1:- A subfamily m of fuzzy sets 1. ∈ , 2. If . Then is called fuzzy m-space.
is called an fuzzy m-space on X if m satisfies the following conditions
Proposition 3.2:- Given a fuzzy m – space m, Example 3.2.1:- Let , where
is a classical m – space. , . Then m is a fuzzy m space.
Definition 3.3:- A fuzzy set .
is called an m-fuzzy open set if
.
is called and m-closed set if
Definition 3.4:- A sub collection g on is called fuzzy grill on X if G satisfies following conditions. 1. G 2. G, G where 3. G G G. A fuzzy m- space with a fuzzy grill G on X is called fuzzy grill m-space and is denoted by Example 3.4.1:- Let
. Let G consist
G).
and all fuzzy sets G of X s.t.
,
G is a fuzzy grill. Rashmi Singh,IJRIT
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IJRIT International Journal of Research in Information Technology, Volume 2, Issue 6, June 2014, Pg: 01-06
:- A is called a of B if BqU for some m –fuzzy open set U in X with Definition 3.5:addition if A itself is m- fuzzy open then it is called an m-open of . We denote the collection of all m open by
, In of
Definition 3.6:-
Definition 3.7:-
Definition 3.8:- m-adherence point of fuzzy set A is
if every m –open
of
is quasi- coincident with A.
Theorem 3.9:is union of all m-adherence points of A. Proof:First we will prove that is just the set of all the m-adherence points if A. Suppose .For all , if Then ⇒ , for each , But So ,by definition we have CONTRADICTION. for some is adherent point of A. Conversely, if is an m-adherent So point of A . Let but and for some , Since is m-adherent point of A and for some 1>1 , CONTRADICTION. So . So is just set of all m- adherent point of A. We know that . So is union of all m-adherence point of A. Definition 3.10:- Fuzzy – Operator Let ( be a fuzzy m-space and G be a fuzzy grill on X. A mapping : denoted by (A) = (A) for all . Where A is a fuzzy set,. is called operator associated with fuzzy grill G and fuzzy { m – structure on Proposition 3.11:- Let ( G ) be a fuzzy grill m-space. (i) If G be a fuzzy grill and G,Then (ii) ( = (iii) For , if ( ( (iv) If and are two fuzzy grills on fuzzy m-space (v) For Proof:(i) Let (ii) G, (iii) , So (iv) Let
,
.Then
for
.
= ⇒ for all
( (A) (B)
with
,
for all
(
, Then for all
G, But
),
G, , Thus for all
G,Now (
),
G
),
(v) Let
Proposition 3.12:- Let ( Rashmi Singh,IJRIT
, Conversely we will prove that ( ) s.t. G and G,Thus ( )
G, Then . So
(
)⇒ ( B).
G) be fuzzy grill m-space, then
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IJRIT International Journal of Research in Information Technology, Volume 2, Issue 6, June 2014, Pg: 01-06
(i) for G (ii) ) for . (iii) ( , for (iv) is m-closed set for Proof: - For G (i) ( A) (ii) Let , Then an m- open G⇒ (iii) Let ( and ( and (iv) (
(
U of ,( (
)
in
i.e. ,then
⇒
, By definition of
Definition 3.15:- Let ( on where for ,
=
G ) be fuzzy grill m-space, then
are two fuzzy grills on
( A))
,
by
is fuzzy topology
, Then
with
= , but
is
(
(
,
we define
and
, Let ( A)
G
Definition 3.13:-Let G be a fuzzy grill on fuzzy m-space for Proposition 3.14:is a Kuratowski closure operator
Theorem 3.16:- if Proof:Let
for each
So
(
⇒
.
–closed
Proposition 3.17:- Let G be fuzzy grill on Proof:- Since G , Then
and B
G. Then
Proposition 3.18:- For any fuzzy subset A of a fuzzy m- space Proof: - For (
is closed in ( , B is
).
and fuzzy grill G on
, is
Remark 3.19:- Let ( G ) be fuzzy grill m-fuzzy grill space for two fuzzy grills may not be a fuzzy grill on but Example: - Let { , }, { , , .Where , is fuzzy m- space. Let consist of and all fuzzy sets of is a fuzzy grill. Let consist of and all fuzzy sets of , is a fuzzy grill. Let C, D are two fuzzy sets , but . Let , nor Theorem 3.20:- Let ( G ) be fuzzy grill m-space and and Then is a fuzzy grill m-space. Proof: - For (i) , (ii) A , ⇒ = , Take
Rashmi Singh,IJRIT
,
be two fuzzy grill,
,
- closed. is - closed. ,
and
– closed.
is fuzzy grill on . Then . Then Then but
, but neither
=
,
and at least one is contained in other. Suppose, Let Now and
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IJRIT International Journal of Research in Information Technology, Volume 2, Issue 6, June 2014, Pg: 01-06
,
(iii) Suppose,
, =
,
,
and
, Theorem 3.21:- Let (a) (b) Proof:- For (a) U ( .Thus (b) Let
and
Λ
be two fuzzy grill on fuzzy m- space (
, Now Then U (A) . Thus (A) Then and and G ≤ G and G = G Λ G
)
and
be two fuzzy grills on (
), then for any fuzzy set
where
.
≰
Conversely Let ( ) s.t. (
and ), So that .
⇒ for each ,
Example 3.21.1:- Let where ( is fuzzy m-space. Let consist of all fuzzy sets of X s.t. of all fuzzy sets in X s.t. and also a fuzzy grill. We see that since for each , Hence ⇒ Hence (A) . Theorem 3.22:- Let
or or
So either
Let
, and are two fuzzy grills on
. Let (
Then consist is ,
. Let then
), , We take
.Then .
≰
=
, then
⇒
Proof: - Since Conversely, Let ⇒
⇒
Thus
and
, Now we will prove
⇒ ⇒
and ⇒
⇒ ⇒
, Since
, , Now
,
⇒
,
,
Thus the trivial F- grill is
Example 3.22.1:- In
{
, G ) be fuzzy grill Theorem 3.23:- Let ( Proof:- Here two cases will arise for any Case 1:- If Case 2 :-
-space for , for
, We take G = , Thus
{
Then we get .
.
, In both case
Theorem 3.24:- Let ( G ) be fuzzy grill -space and G be a fuzzy grill on . Then G} is fuzzy open base for Proof:- First we show that (G, is a sub collection of , Let (G, We will prove that ( -U) - , Let a fuzzy point s.t. ( - (i) But ( - (ii)
Rashmi Singh,IJRIT
(G,
=
and G.
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IJRIT International Journal of Research in Information Technology, Volume 2, Issue 6, June 2014, Pg: 01-06
(i) implies that for each ( >( (Since in that case
)
(
G - (iii) ,By (ii) + In fact is not possible. , So and , Then Thus ) , By (ii) G, by above theore G CONTRADICTION to G, Thus (G, is sub collection of . Next , Let be any fuzzy point in and U be an - open of in ( , ), Then by def. of ∃a B s.t. and is - closed and ( ⇒ ( , ( a -open V of in s.t. ( G, By (a) G , We will show that q , Since and , We have , Then if , we have .Thus we have for each fuzzy point in ) and for each open of in , ) , (G, and ( , , (G, is a base of . G,
- (a),
G
Corollary 3.24.1:- For any fuzzy -space (G, G) be fuzzy grill -space Example 3.24.2:- Let ( Take G = { } Then In Fact for fuzzy basic m-open set , (G, We have
with
,
G in
4. Fuzzy m-space suitable with fuzzy grill in fuzzy m-space Definition 4.1:- Let G be a fuzzy grill on a fuzzy m-space ( for every , if corresponding to each , , we define
Definition 4.2:- For every Proposition 4.3:- For any (i) (ii) ( )
={
). Then ) /
}
Theorem 4.4- For a fuzzy grill G on . The following are equivalent (a) is suitable for fuzzy grill G (b) For , G (c) for (d) For every set in , if A contains no non empty fuzzy set with (e) For fuzzy set in , ∉ G
Proof:(a) ⇒ (b) Let, . Hence suitable for
for any given fuzzy set By definition of G,
in
Then
. Now for each fuzzy point some U ) s.t.
G
, we have G. Since m
is
G.
( ) (A) ⇒ (A). So ( ) = λ.Then ( ) (A), CONTRADICTION , ( ) (c) ⇒ (d) Let which contains no non empty fuzzy set s.t. ( ) ( (A)), By (c) ∉ G ( ) ( , But G
Rashmi Singh,IJRIT
–space is said to be suitable for fuzzy grill G if G, Then ∉ G.
( ) ⇒ ( ) , We know that , We have = (
.But
λ, By (b)
⇒ G. (
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IJRIT International Journal of Research in Information Technology, Volume 2, Issue 6, June 2014, Pg: 01-06
, ={ , We claim that does not contain any non empty fuzzy . Then by ∉ G if possible, Let be non empty fuzzy set contained in s.t. Let ≤ A but But ⇒ ( ) ( )= , CONTRADICTION. , Then ∉ G for - closed set. (c) ⇒ (e) ∉ G for (e) ⇒ (a) Let be any fuzzy set in with the property that for every fuzzy point A, G (B) , , G, Let but , As , , , Then G, G (d) ⇒ ( c) Let
Theorem 4.5:- Let ( , be a fuzzy space, then following are equivalent (a) ( b) For ⇒ , ( )= (c) For (d) For , Proof:(a) ⇒ (b) follows from Theorem 4.4. (b) ⇒ (c) Let , using Theorem 4.3 ( ) ( ) . , then by proposition 4.3, (c) ⇒ (d) Let (d) ⇒ (a) Let
,
Theorem 4.6:- If
(
Proof:- For .But
, (
≤
space is suitable for fuzzy grill G
( ) .
and by (d)
space of
set
is suitable for G, then
is idempotent operator i.e.
(
for
( m is suitable for G .So
(
Theorem 4.7:- Let be a m-space and G be a fuzzy grill on s.t. m is suitable for G. Then a fuzzy set in - closed set if it can be expressed as a union of a fuzzy set which is closed in (X, m) and a fuzzy set which is not in G. Proof:- Let then G. ∉ G and is closed. Conversely Let where is closed, ∉ G⇒ = . . is – closed be fuzzy m –space and G be fuzzy grill s.t. is suitable for G . Let = denote fuzzy is –space , So is only - open of every in .Thus for , G G. Thus for G, , for every fuzzy set ∉ G. Thus V if G , if ∉ G = G } , clearly . Since for , ∉G. ( is - closed for any , we claim that . Clearly m , . Conversely, let where is –closed and ∉ G, U Λ( ) where ( ) is –open and is – open. So . So
Example 4.8:- Let indiscrete topology on
=(
Theorem 4.9:- Let G be a fuzzy grill on with m is suitable for G. Then (G, is fuzzy topology on . Hence (G, m) = Proof:(G, ( Proved), on the other hand , Let be - closed .Then , So By definition of , ( by (a).Since is suitable for G ⇒ ∉ G. Thus V .Where is –closed and ∉ G. We can write - by (a) [ ] [ . Where , ∉G, (G, . So (G, Rashmi Singh,IJRIT
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References [1] B. Roy, M. N. Mukherjee, On a typical topology induced by grill, Soochow J. Math. 33,4 2007, 771-786. [2] C. L. Chang, Fuzzy topological space, J. Math. Anal. Appl. 24, 1968, 182-190. [3] G. Choquet, Sur les notions de felters et de grille, C.R. Acad. Sci. Paris, 224, 1947, 171-173. [4] K.K. Azad, Fuzzy grills and a characterization of fuzzy proximity, J. Math. Anal. Appl., 79, 1981, 13-17. [5] Khare, M., Singh, R.: L-guilds and binary L-merotopies, Novi Sad J. Math., 36, 2, 2006, 57–64. [6] Khare, M., Singh, R.: L-contiguities and their order structure, Fuzzy Sets and Systems,158, 4, 2007, 399–408. [7] Khare, M., Singh, R.: Complete ξ-grills and (L, n)-merotopies, Fuzzy Sets and Systems, 159, 5,2008, 620–628. [8] L.A. Zadeh, Fuzzy sets, Inform. Control, 8, 1965, 338-353. [9] M. N.Mukherjee and Sumita Das, Fuzzy Grills and induced fuzzy topology, MATEMATИ4KИ BECHИK, 62, 2010, 285-297. [10] P. Srivastava, R. L. Gupta, Fuzzy proximity structures and fuzzy ultrafilters, J. Math. Anal. Appl., 94, 2, 1983, 297311. [11] Shyamapada Modak, Operators on grill m-space, Bol. Soc. Paran. Mat. , 31, 2, 2013, 101-107.
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