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ISSN 2320-5407
International Journal of Advanced Research (2015), Volume 3, Issue 7, 1188-1191
Journal homepage: http://www.journalijar.com
INTERNATIONAL JOURNAL
OF ADVANCED RESEARCH
RESEARCH ARTICLE
∗ −Closed sets in Fine-Topological space
1
Powar P. L., 2Rajak K., 3Tiwari V.
1: Professor, Department of Mathematics and Computer Science, R.D.V.V., Jabalpur
2: Asst. Prof., Department of Mathematics, St. Aloysius College (Auto), Jabalpur
3: Department of Mathematics and Computer Science, R.D.V.V., Jabalpur.
Manuscript Info
Abstract
Manuscript History:
The aim of this paper is to introduce a new class of closed sets called finepre-g*-closed (fpg*-closed sets) using  − open sets in fine-topological
space. Also we study the some property of fpg*-closed sets in finetopological space.
Received: 14 May 2015
Final Accepted: 22 June 2015
Published Online: July 2015
Key words: fpg*-clsosed sets,
fpg*-open sets, fine-open sets.
*Corresponding Author
Powar P. L
Copy Right, IJAR, 2015,. All rights reserved
1. INTRODUCTION
In 1969, Levine [10] gives the concept and properties of generalized closed (briey g-closed) set and the
complement of g-closed set is said to be g-open set. Later, Noiri et al. [11], Dontechev [5], Gnanambal [6], Arya and
Nour [1], Bhattacharya and Lahiri [4], Maki et al. [12, 13] and Sundaram and Sheik John [21] introduced and
studied the concept of gp-closed, gsp-closed, gpr-closed, gs-closed, sg-closed, g-closed, g-closed and -closed
sets and their complements are respectively open sets. Moreover, the notion of -closed [3] sets was introduced by
the present authors.
Powar P. L. and Rajak K. [20], have investigated a special case of generalized topological space called fine
topological space. In this space, they have defined a new class of open sets namely fine-open sets which contains all
 −open sets,  −open sets, semi-open sets, pre-open sets, regular open sets etc. By using these fine-open sets they
have define fine-irresolute mappings which include pre-continuous functions, semi-continuous function,
 −continuouos function,  −continuous functions,  −irresolute functions,  −irresolute functions, etc (cf. [12][16]).
In the present paper, the author have introduce a new class of closed sets called fine-pre-g*-closed (fpg*-closed
sets) using  − open sets in fine-topological space. Also we study the some property of fpg*-closed sets in finetopological space.
2. Preliminaries
Throughout this paper, (, ) and (, ) means topological spaces on which no separation axioms are assumed.
For a subset A of a space X the closure and interior of A with respect to  are denoted by () and   . We use
the following definitions:
Definition 2.1 A subset A of a space (, ) is called
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ISSN 2320-5407
International Journal of Advanced Research (2015), Volume 3, Issue 7, 1188-1191
1) Semi-open if  ⊆    (cf. [11]).
2)  −open if ⊆ ((())) (cf. [8]).
3)  −open if  ⊆ ((())) (cf. [3]).
4) Pre-open if  ⊆ (()) (cf. [1]).
The complements of  −open sets,  −open sets, semi-open sets, pre-open sets are called  −closed sets,
 −closed sets, semi-closed sets, pre-closed sets
Definition 2.2 A subset A of ,  is said to be generalized closed (briefly g−closed) if cl(A) ⊆ U whenever A ⊆
U and U is open in (X, ) (cf. [7]).
Definition 2.3 A subset A of ,  is said to be weakly closed set (−closed) set if cl(A) ⊆ U whenever A ⊆ U and
U is semi-open in (X, ). The complement of  −closed set is called  −open in ,  (cf. [7]).
Definition 2.4 A subset A of ,  is said to be  − closed if   ⊆  whenever  ⊆  and U is  −open set
in ,  (cf. [7]).
Definition 2.5 A subset A of ,  is said to be pre-g*− closed (briefly ∗ −closed) if   ⊆  whenever
 ⊆  and U is  −open set in ,  . The complement of ∗ −closed set is called ∗ −open in ,  (cf. [7]).
Definition 2.6 Let (X, τ) be a topological space we define
τ ( ) =  (say) = { (≠X) :  ∩  = , for  ∈  and  = , , for some α ∈ J, where J is the index set.}
Now, we define
 = {, ,∪ ∈ { }}
The above collection  of subsets of X is called the fine collection of subsets of X and (X, τ,  ) is said to be the
fine space X generated by the topology τ on X (cf. [20]).
Definition 2.7 A subset U of a fine space X is said to be a fine-open set of X, if U belongs to the collection  and
the complement of every fine-open sets of X is called the fine-closed sets of X and we denote the collection by
 (cf. [20]).
Definition 2.8 Let A be a subset of a fine space X, we say that a point x ∈ X is a fine limit point of A if every fineopen set of X containing x must contains at least one point of A other than x (cf. [20]).
Definition 2.9 Let A be the subset of a fine space X, the fine interior of A is defined as the union of all fine-open sets
contained in the set A i.e. the largest fine-open set contained in the set A and is denoted by  (cf. [20]).
Definition 2.10 Let A be the subset of a fine space X, the fine closure of A is defined as the intersection of all fineclosed sets containing the set A i.e. the smallest fine-closed set containing the set A and is denoted by  (cf. [20]).
Definition 2.11 A function f : (X, τ,  ) → (Y, τ ′, ′ ) is called fine-irresolute (or f-irresolute) if  −1 ( ) is fineopen in X for every fine-open set V of Y (cf. [20]).
3. ∗ −closed sets in fine-topological spaces
In this section, we introduce ∗ −closed sets in fine-topological space and study some of their properties.
Definition 3.1 A subset A of fine-topological space , ,  is said to be fine-generalized closed (briefly
fg−closed) if  () ⊆  whenever A ⊆ U and U is fine-open in (X, ,  ).
Example 3.2 Let  = , ,  and  = , ,  , ,  ,  = {, ,  ,  , ,  ,
,  , {, }} ,  = , , ,  , ,  ,  ,  ,  . It may be easily check that, the set  is fg-closed.
Definition 3.3 A subset A of fine-topological , ,  is said to be fine-weakly closed set (−closed) set if
 () ⊆  whenever A ⊆ U and U is fine-semi-open in (X, ,  ). The complement of f −closed set is called
f −open in , ,  .
Example 3.4 Let  = , ,  and  = , , ,  ,  = {, ,  ,  , ,  ,
,  , {, }} ,  = , , ,  , ,  ,  ,  ,  . It may be easily check that, the set ,  is -closed.
Definition 3.5 A subset A of fine-topological , ,  is said to be  − closed if   ⊆  whenever  ⊆ 
and U is f −open set in , ,  .
Example 3.6 Let  = , ,  and  = , , ,  ,  = {, ,  ,  , ,  ,
,  , {, }} ,  = , , ,  , ,  ,  ,  ,  . It may be easily check that, the set ,  is -closed.
Definition 3.7 A subset A of fine-topological , ,  is said to be fine pre-g*− closed (briefly ∗ −closed) if
  ⊆  whenever  ⊆  and U is  −open set in , ,  . The complement of ∗ −closed set is called
∗ −open in ,  .
Example 3.8 Let  = , ,  and  = , , ,  ,  = {, ,  ,  , ,  ,
,  , {, }} ,  = , , ,  , ,  ,  ,  ,  . It may be easily check that, the sets  , ,  is  ∗-closed.
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Remark 3.9: Every f −closed set,  − closed set,  −closed set and ∗ −closed sets are fine-closed.
Theorem 3.10 Every fine-closed (resp. fine-pre-closed) set is fpg*-closed.
Proof: Let A be any fine-closed and G be any  − open set containing A in (, ,  ). Since, A is fine-closed
  =  (resp.   = ), so   ⊆ . Hence, A is fpg*-closed in , ,  .
Theorem 3.11 Every  −closed set is fpg*-closed set.
Proof: Let A be  closed and G be  −open set in (X, ,  ) such that  ⊆ . Since A is  -closed,
   = . But    ⊆  () is always true. Thus,    ⊆ . Hence A is  ∗-closed in (X,,  ), the proof
follows.
Theorem 3.12 A fine-regular open fpg*-closed set is f-pre-closed and hence f-clopen.
Proof. Let A be regular open, fpg*-closed. Since fine-regular open set is fine-open ,   ⊆ . This implies A is
f-pre-closed. Since every f-pre-closed (fine-regular) open set is (fine-regular) fine-closed, A is fine-clopen.
4. fpg*-Open Sets
In this section, the notion of fpg*-open set is defined and some of its basic properties are studied.
Definition 4.1 A subset A in X is called fpg*-open in (X,,  ) if  −  is ∗-closed (, ,  )
Example 4.2 Let  = , ,  and  = , , ,  ,  = {, ,  ,  , ,  ,
,  , {, }} ,  = , , ,  , ,  ,  ,  ,  . It may be easily check that, the sets ,  is ∗-open.
Theorem 4.3 A set A is fpg*-open in (X,,  ) if and only if  ⊆  () whenever F is - closed in (X,,  )
and  ⊆ .
Proof. Suppose  ⊆  () where F is f-closed and  ⊆ . Let  −  ⊆  where G is f- open in
(X,,  ). Then  ⊆  −  and  −  ⊆  (). Thus  −  is ∗-closed in (X,,  ). Hence A is
∗ −open in (X,,  ).
Conversely, suppose that A is ∗-open,  ⊆  and F is f −closed in (X,,  ). Then  −  is -open and
 −  ⊆  − . Therefore   −  ⊆  − . But   −  =  −  (). Hence  ⊆  ().
5. Conclusion
By using the concepts of fpg*-closed sets on fine-topological space, we may define a generalized form of
continuity in terms of fpg*-closed sets. Also, by defining some irresolute maps, the more general form of
homeomorphism can be studied which is widely used in quantum physics.
6. Acknowledgement
Authors of the paper would like to thank UGC for extending the financial support under DSA- I
programme vide letter number F510/3/DSA/2013(SAPI) dated Sept. 10, 2013 to the Department of Mathematics
and Computer Science, R.D.V.V, Jabalpur.
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