Pairwise Connectedness In Soft BiČech Closure Spaces

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Journal of Mathematics and InformaticsVol. 6, 2016, 1-6ISSN: 2349-0632 (P), 2349-0640 (online)Published 7 September 2016www.researchmathsci.orgJournal ofPairwise Connectedness in soft biČech Closure SpacesR. Gowri1 and G. Jegadeesan21Department of Mathematics, Govt. College for Women’s (A)Kumbakonam-612 001, Tamil Nadu, India. Email: gowrigck@rediffmail.com2Department of Mathematics, Anjalai Ammal Mahalingam CollegeKovilvenni-614 403 Tamil Nadu, India. Email: jega0548@yahoo.co.inReceived 23 August 2016; accepted 30 August 2016Abstract. The aim of the present paper is to study the concept of pairwise connectednessin biČech closure spaces through the parameterization tool which is introduced byMolodtsov.Keywords: Pairwise soft separated sets, pairwise connectedness, pairwise feeblydisconnectedness.AMS Mathematics Subject Classification (2010): 54A05, 54B051. IntroductionČech [1] introduced the concept of closure spaces and developed some properties ofconnected spaces in closure spaces. According to him, a subset A of a closure space X issaid to be connected in X is said to be connected in X if A is not the union of two nonempty Semi-Separated Subsets of X.Plastria studied [2] connectedness and local connectedness of simple extensions.Rao and Gowri [3] studied pairwise connectedness in biČech closure spaces.Gowri and Jegadeesan [7,8,9,10] introduced separation axioms in soft Čech closurespaces, soft biČech closure spaces and studied the concept of connectedness in fuzzy andsoft Čech closure spaces.In 1999, Molodtsov [4] introduced the notion of soft set to deal with problems ofincomplete information. Later, he applied this theory to several directions [5] and [6].In this paper, we introduced and exhibit some results of pairwise connectednessin soft biČech closure spaces.2. PreliminariesIn this section, we recall the basic definitions of soft biČech closure space.Definition 2.1. [9] Let X be an initial universe set, A be a set of parameters. Then thefunction : and : defined from a soft power set ( ) to itself over X is called Čech Closure operators if it satisfies the followingaxioms:(C1) ( ) and ( ) .(C2) ( )and ( ).(C3) ( ) ( ) ( ) and ( ) ( ) ( ).1

R. Gowri and G. JegadeesanThen (X, , , A) or ( , , ) is called a soft biČech closure space.Definition 2.2. [9] A soft subset of a soft biČech closure space ( , , ) is said tobe soft , -closed if ( ) , 1,2. Clearly, is a soft closed subset of a softbiČech closure space ( , , ) if and only if is both soft closed subset of ( , )and ( , ).Let be a soft closed subset of a soft biČech closure space ( , , ). The followingconditions are equivalent.1. ( ) .2. ( ) and ( ) .Definition 2.3. [9] A soft subset of a soft biČech closure space (( , , )) is said tobe soft , -open if , 1,2.Definition 2.4. [9] A soft set !"# % ( ), 1,2 with respect to the closure operator isdefined as !"# % ( ) ( ) ' ( , 1,2. Here .Definition 2.5. [9] A soft subset in a soft biČech closure space ( , , ) is calledsoft , neighbourhood of ) * ) !"# %,-,. ( ).Definition 2.6. [9] If ( , , ) be a soft biČech closure space, then the associate softbitopology on is / 0 : ( ) , 1,21.Definition 2.7. [9] Let ( , , ) be a soft biČech closure space. A soft biČech closurespace (2 , , ) is called a soft subspace of ( , , ) if 2 F5 and ( ) ( ) 2 , 1,2, for each soft subset G5 .3. Pairwise connectednessIn this section, we introduce pairwise soft separated sets and discuss the pairwiseconnectedness in soft biČech closure space.Definition 3.1. Two non-empty soft subsets U5 and V5 of a soft biČech closurespace ( , , ) are said to be pairwise soft separated if and only if U5 V5 5and U5 V5 5.Remark 3.2. In other words, two non-empty U5 and V5 of a soft biČechclosure space ( , , ) are said to be pairwise soft separated if and only if(U5 V5 ) ( U5 V5 ) 5.Theorem 3.3. In a soft biČech closure space ( , , ), every soft subsets of pairwisesoft separated sets are also pairwise soft separated.Proof. Let ( , , ) be a soft biČech closure space. Let U5 and V5 are pairwise softseparated sets. Let 2 ?"@ A . Therefore, U5 V5 5 and U5 V5 5 . (1)2

Pairwise Connectedness in soft biČech closure spacesSince, 2 2 2 H5 A 2 H5 2 H5 5 by (1) 2 H5 5 .Since, A A A G5 2 A G5 A G5 5 by (1) A G5 5 .Hence, U5 and V5 are also pairwise soft separated.Theorem 3.4. Let (G5 , , ) be a soft subspace of a soft biČech closure space( , , ) and G)# , V5 G5, then U5 and V5 are pairwise soft separated in if andonly if U5 and V5 are pairwise soft separated in G5 .Proof. Let ( , , ) be a soft biČech closure space and (G5, , ) be a soft subspaceof ( , , ). Let U5, V5 G5. Assume that, U5 and V5 are pairwise soft separated in implies that U5 V5 5 and U5 V5 5 . That is, (U5 V5 ) ( U5 V5 ) 5.Now,(U5 V5 ) ( U5 V5 ) U5 ( V5 G5 ) ( U5 G5 ) V5 (U5 G5 V5 ) ( U5 G5 V5 ) (U5 V5 ) ( U5 V5 ) 5 .Therefore, U5 and V5 are pairwise soft separated in F5 if and only if U5 and V5 arepairwise soft separated in G5 .Definition 3.5. A soft biČech closure space ( , , ) is said to be pairwisedisconnected if it can be written as two disjoint non-empty soft subsets U5 and V5 suchthat U5 V5 5 and U5 V5 F5 .Definition 3.6. A soft biČech closure space( , , ) is said to be pairwise connected ifit is not pairwise disconnected.Example 3.7. Let the initial universe set HI , I J and K HL , L , LM Jbe the parameters. Let N HL , L J Kand H(L , HI , I J), (L , HI , I J)J. Then are, H(L , HI J)J, H(L , HI J)J, M H(L , HI , I J)J, O H(L , HI J)J, P H(L , HI J)J, Q H(L , HI , I J)J, R H(L , HI J), (L , HI J)J, S H(L , HI J), (L , HI J)J, T H(L , HI J), (L , HI J)J, U H(L , HI J), (L , HI J)J, H(L , HI J), (L , HI , I J)J, H(L , HI J), (L , HI , I J)J, M H(L , HI , I J), (L , HI J)J, O H(L , HI , I J), (L , HI J)J, P , Q .An operator : ( ) ( ) is defined from soft power set ( ) to itself over Xas follows. ( ) , ( ) , ( M ) M , ( O ) O , ( P ) P , ( Q ) Q , ( R ) R , ( S ) S , ( T ) T , ( U ) U , ( ) , ( ) , ( M ) M , ( O ) O ,3

R. Gowri and G. Jegadeesan ( ) , ( ) .An operator : ( ) ( ) is defined from soft power set ( ) to itself over Xas follows. ( ) ( ) ( M ) M , ( O ) ( Q ) Q , ( P ) P , ( R ) ( T ) ( ) ( ) ( M ) ( ) , ( ) , ( S ) ( U ) ( O ) O .Taking, O ?"@ M , and U5 V5 .Therefore, the soft biČech closure space ( , , ) is pairwise disconnected.Example 3.8. Let us consider the soft subsets of that are given in example 3.7.An operator : ( ) ( ) is defined from soft power set ( ) to itself over Xas follows. ( ) ( M ) ( O ) ( R ) ( T ) ( M ) M , ( Q ) ( S ) ( ) ( ) ( O ) ( ) , ( ) . ( ) T , ( U ) , ( P ) P .An operator : ( ) ( ) is defined from soft power set ( ) to itself over Xas follows. ( ) ( R ) ( S ) ( ) , ( O ) ( P ) ( Q ) Q , ( ) U , ( T ) ( U ) ( ) , ( ) , ( M ) ( M ) ( O ) ( ) .Here, the soft biČech closure space ( , , ) is pairwise connected.Remark 3.9. The following example shows that pairwise connectedness in soft biČechclosure space does not preserves hereditary property.Example 3.10. In example 3.8., the soft biČech closure space ( , , ) is pairwiseconnected. Consider (G5 , , ) be the soft subspace of such that 2 H(L , HI , I J)J. Taking, H(L , HI J)J ?"@ H(L , HI J)J, and 2 . Therefore, the soft biČech closure subspace (G5, , ) ispairwise disconnected.Theorem 3.11. Pairwise connectedness in soft bitopological space ( , / , / ) need notimply that the soft biČech closure space ( , , ) is pairwise connected.Proof. Let us consider the soft subsets of that are given in example 3.7. An operator ( ) , ( ) ( T ) , ( O ) O , ( P ) ( S ) O , ( R ) R , ( M ) ( Q ) ( U ) ( ) ( ) ( M ) ( O ) ( ) , ( ) .An operator : ( ) ( ) is defined from soft power set ( ) toitself over X as follows. ( ) ( P ) S , ( ) M , ( O ) O , ( R ) R , ( Q ) ( S ) ( ) , ( M ) ( T ) ( M ) M , ( U ) O , ( ) ( O ) ( ) , ( ) .4

Pairwise Connectedness in soft biČech closure spacesHere, the two non empty disjoint soft subsets H(L , HI J)J,?"@ H(L , HI J)J,satisfies and .Therefore, the soft biČech closure space ( , , ) is pairwise disconnected. But, it’sassociated soft bitopological space ( , / , / ) is / H , U , , O , J and/ H , , P , U , O , J.Now, / VG( ) / VG( ) H(L , HI J)J H(L , HI J)J H(L , HI J)J H(L , HI J)J . Therefore, ( , / , / ) is pairwise connected.Theorem 3.12. If soft biČech closure space is pairwise disconnected such that / and let 2 be a pairwise connected soft subset of then 2 neednot to be holds the following conditions ( )2 ( )2 .Proof. Let us consider the soft subsets of that are given in example 3.7. An operator : ( ) ( ) is defined from soft power set ( ) to itself over X as follows. ( ) ( P ) S , ( ) M , ( O ) O , ( R ) R , ( Q ) ( S ) ( ) , ( M ) ( T ) ( M ) M , ( U ) O , ( ) ( O ) ( ) , ( ) .An operator : ( ) ( ) is defined from soft power set ( ) to itself over Xas follows. ( ) ( M ) ( O ) ( R ) ( T ) ( M ) M , ( Q ) ( S ) ( ) ( ) ( O ) ( ) , ( ) . ( ) T , ( U ) , ( P ) P .Taking, ?"@ P then we get, / .Here, the soft biČech closure space ( , , ) is pairwise disconnected. Let 2 R be the pairwise connected soft subset of . Clearly, 2 does not lie entirely within either YZ .Theorem 3.13. If the soft bitopological space ( , / , / ) is pairwise disconnected thenthe soft biČech closure space ( , , ) is also pairwise disconnected.Proof. Let the soft bitopological space ( , / , / ) is pairwise disconnected,implies that it is the union of two non empty disjoint soft subsets U5 and V5 such that U5 τ VG(V5 ) τ VG(U5 ) V5 5. Since, , U5 τ[ , VG(U5 ) forevery U5 F5 and τ VG(U5) τ VG(V5 ) 5 then U5 V5 5 . Since,U5 V5 F5 , U5 U5 and V5 V5 implies that U5 V5 U5 V5 ,F5 U5 V5 . But, U5 V5 . Therefore, U5 V5 F5 .Hence, ( , , ) is also pairwise disconnected.Definition 3.14. A soft biČech closure space ( , , ) is said to be pairwise feeblydisconnected if it can be written as two non-empty disjoint soft subsets U5and V5 suchthat U5 V5 and U5 V5 .Result 3.15. Every pairwise disconnected soft biČech closure space ( , , ) ispairwise feebly disconnected but the following example shows that the converse is nottrue.5

R. Gowri and G. JegadeesanExample 3.16. In example 3.8 Consider, S H(L , HI J), (L , HI J)J and H(L , HI J)J. Which satisfies the condition U5 V5 U5 V5 . Therefore, the soft biČech closure space ( , , ) ispairwise feebly disconnected. But, the soft biČech closure space ( , , ) is pairwiseconnected.REFERENCES1. E.Čech, Topological spaces, Interscience publishers, John Wiley and Sons, NewYork(1966).2. F.Plastria, Connectedness and local connectedness of simple extensions, Bull. Soc.Math. Belg., 28 (1976) 43-51.3. K.Chandrasekhara Rao and R.Gowri, Pairwise connectedness in biČech closurespaces, Antartica J. Math., 5(1) (2008) 43-50.4. D.A.Molodtsov, Soft set theory- first results, Comput Math. Appl., 37 (1999) 19-31.5. D.A.Molodtsov, The description of a dependence with the help of soft sets, J.Comput. Sys. Sc. Int., 40 (2001) 977-984.6. D.A.Molodtsov, The theory of soft sets (in Russian), URSS publishers, Moscow(2004).7. R.Gowri and G.Jegadeesan, Connectedness in fuzzy Čech closure spaces, Asian. J.Current Engg and Math., (2) (2013) 326-328.8. R.Gowri and G.Jegadeesan, On soft Čech closure spaces, Int. J. Math. Trends andTechnology, 9(2) (2014) 122-127.9. R.Gowri and G.Jegadeesan, On soft biČech closure spaces, Int. J. Math. Archive, 5(11) (2014) 99-105.10. R.Gowri and G.Jegadeesan, Connectedness in soft Čech closure spaces, Annals ofPure and Applied Mathematics, 11(1) (2016) 115-122.6

Therefore, (ˆ ,/ ,/ ) is pairwise connected. Theorem 3.12. If soft biČech closure space is pairwise disconnected such that ˆ / and let 2 be a pairwise connected soft subset of ˆ then 2 need not to be holds the following conditions ( )2 ( )2 . Proof.

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