On topologies induced by the soft topology

Tamsui Oxford Journal of Information and Mathematical Sciences 31(1) (2017) 49-59 Aletheia University On topologies induced by the soft topology Evanzalin Ebenanjar P. y Research scholar, Department of Mathematics, Karunya University, Coimbatore-641114, India. and Thangavelu P. z Ramanujam Centre for Mathematical Sciences, Thiruppuvanam-630611, India. Received January 23, 2016, Accepted June 10, 2016. Abstract Molodtsov introduced soft sets to deal with uncertainty. Shabir introduced soft topological spaces and established that every soft topological space induce a parameterized family of topological spaces. In this paper some concepts in soft topological spaces are characterized by using the analogous concepts in the parameterized family of topological spaces induced by the soft topology. Keywords and Phrases: Soft sets, soft topology, soft function, soft continuous, soft locally nite, parameterized family of topological spaces. 2000 Mathematics Subject Classication. Primary 06D72, 54A10. y E-mail: evanzalin86@yahoo.com z Corresponding author. E-mail: ptvelu12@gmail.com

50 Thangavelu P. and Evanzalin Ebenanjar P. 1. Introduction The real life problems in Social sciences, Engineering, Medical sciences, Economics and Environment deal with the unpredictability and it is imprecise in nature. There are several theories in the literature such as Fuzzy set theory[16], Rough set theory[12], Interval Mathematics[2] etc dealing with uncertainties, but they have their own limitations. A new Mathematical tool dealing with uncertainty is introduced by Molodtsov[9] in 1999. Molodtsov initiated the theory of soft sets for modeling vagueness and uncertainty. Maji et.al[7, 8] studied the basic operations on soft sets and applied soft sets in decision making problems. Kharal et.al.[6] studied soft functions and their application in Medical expert system. Shabir et.al.[10] initiated the study of soft topological spaces. Moreover, theoretical studies of soft topological spaces have also been studied in [14],[3],[17],[15],[4]. Soft continuous functions are studied in [17]. Soft connectedness and soft Hausdor spaces were introduced in [13]. In this paper some concepts in soft topological spaces are characterized by using the analogous concepts in the parametrized family of topological spaces induced by the soft topology. 2. Preliminaries Throughout this paper X,Y are universal sets and E,K are parameter spaces. Denition 2.1 ([9]). A pair (F; E) is called a soft set over X where F : E! 2 X is a mapping. S(X; E) denotes the collection of all soft sets over X with parameter space E. We denote (F; E) by e F in which case we write e F = f(e; F (e)) : e 2 Eg: In some occasions, we use e F (e) for F (e). Denition 2.2 ([8]). For any two soft sets e F and e G in S(X; E), e F is a soft subset of e G if F (e) G(e) for all e 2 E. If e F is a soft subset of e G then we write e F e G.

On topologies induced by the soft topology 51 ef and e G are equal if and only if F (e) = G(e) for all e 2 E. That is e F = e G if ef e G and e G e F. Denition 2.3 ([8]). (i) ë = f(e; ) : e 2 Eg = f(e; (e)) : e 2 Eg = ( ; E). (ii) e X = f(e; X) : e 2 Eg = f(e; X(e)) : e 2 Eg = (X; E). Denition 2.4 ([8]). The union of two soft sets e F and e G over X is dened as e F [ e G = (F [ G; E) where (F [ G)(e) = F (e) [ G(e) for all e 2 E. Denition 2.5 ([1]). The intersection of two soft sets e F and e G over X is dened as e F \ e G = (F \ G; E) where (F \ G)(e) = F (e) \ G(e) for all e 2 E: If f e F : 2 g is a collection of soft sets in S(X; E) then the arbitrary union and the arbitrary intersection of soft sets are dened below. S f e F : 2 g = ( S ff : 2 g; E) and T f e F : 2 g = ( T ff : 2 g; E) where ( S ff : 2 g)(e) = S ff (e) : 2 g and ( T ff : 2 g)(e) = T ff (e) : 2 g, for all e 2 E. Denition 2.6 ([10]). The complement of a soft set e F is denoted by ( e F ) 0 = (F 0 ; E) (relative complement in the sense of Ifran Ali.et.al([1]) where F 0 : E! 2 X is a mapping given by F 0 (e) = X F (e) for all e 2 E. It is noteworthy to see that with respect to above complement De Morgan's laws hold for soft sets as stated below. Lemma 2.7 ([17]). Let I be an arbitrary index set and f e Fi : i 2 Ig S(X; E). Then ([f e Fi : i 2 Ig) 0 = \f( e Fi ) 0 : i 2 Ig and (\f e Fi : i 2 Ig) 0 = [f( e Fi ) 0 : i 2 Ig. Denition 2.8 ([10]). Let e F be a soft set over X and x 2 X. We say that x 2 e F whenever x 2 F (e) for all e 2 E.

52 Thangavelu P. and Evanzalin Ebenanjar P. Denition 2.9 ([10]). Let Y be a non-empty subset of X. Then e Y denotes the soft set (Y; E) over X if Y (e) = Y for all e 2 E. From denition 2.3(ii) it is easy to see that every soft set in S(X,E) is a soft subset of e X Denition 2.10 ([10]). Let e be a collection of soft subset of e X. Then e is said to be a soft topology on X with parameter space E if (i) ë ; e X 2 e, under nite intersection. (ii) e is closed under arbitrary union, and (iii) e is closed If e is a soft topology on X with a parameter space E then the triplet (X; E; e) is called a soft topological space over X with parameter space E. Identifying (X; E) with e X, ( e X; e) is a soft topological space. The members of e are called soft open sets in (X; E; e). A soft set e F in S(X; E) is soft closed in (X; E; e), if its complement ( e F ) 0 belongs to e. (e) 0 denotes the collection of all soft closed sets in (X; E; e). The soft closure of e F is the soft set, escl( e F ) = \f e G : e G is soft closed and e F e Gg. The soft interior [14] of ef is the soft set, esint( e F ) = [f e O : e O is soft open and e O e F g. It is easy to see that e F is soft open, e F = esint e F and e F is soft closed, e F = escl e F Lemma 2.11 ([10]). Let (X; E; e) be a soft space over X. Then the collection e e = ff (e) : e F 2 eg is a topology on X for each e 2 E. Thus a soft topology on X gives a parameterized family of topologies on X but the converse is not true as shown in Example 1 of [10]. Denition 2.12 ([10]). Let Z be a non-empty subset of X. Then e Z e X. Let (X; E; e) be a soft topological space. Dene e Z = f(z F ; E) : Z F (e) = F (e) \ Z for every e 2 E and e F 2 eg. Then e Z is a soft topology on Z with parameter space E and (Z; E; e Z ) = ( e Z; e Z ) is a soft subspace of (X; E; e). Clearly e Z F = e Z \ e F for all e F 2 e.

On topologies induced by the soft topology 53 Denition 2.13 ([10]). Let (X; E; e) be a soft topological space over X, e G be a soft set over X and x 2 X. Then e G is said to be a soft neighbourhood of x if there exist a soft open set e F such that x 2 e F e G. Denition 2.14 ([11]). A family fa : 2 g of sets in a topological space X is called locally nite(nbd-nite in the sense of [5]) if each point of X has a neighborhood V such that V \ A 6= for at most nitely many indices. 3. Soft locally nite In this section, soft locally nite family of soft sets is introduced and its properties are investigated. The following propositions are useful in sequel. Proposition 3.1 Let (X; E; e) be a soft topological space over X. Let e F 2 S(X; E) and e 2 E. If e F is soft open in (X; E; e) then e F (e) is open in (X; ee ). Conversely if G is open in (X; e e ) then G = e F (e) is soft open in (X; E; e) for some soft open set e F in (X; E; e). Suppose e F 2 e. Then by using the Lemma 2.11, e F (e) 2 ee. Conversely suppose G 2 e e. Again by using the same Lemma 2.11, G = e F (e) for some e F 2 e. Proposition 3.2 Let (X; E; e) be a soft topological space over X. Let e F 2 S(X; E) and e 2 E. If e F is soft closed in (X; E; e) then e F (e) is closed in (X; e e ). Conversely if G is closed in (X; e e ) then G = e F (e) is soft closed in (X; E; e) for some soft closed set e F in (X; E; e) Suppose F e is soft closed. Then ( F e ) 0 is soft open. Then by using the Lemma 2.11, ( F e ) 0(e) is open in (X; e e ). That is X F e (e) is open in (X; ee ). That is F e (e) is closed in (X; ee ) for all e 2 E. Conversely suppose G is closed in (X; e e ). Then G is open in 0 (X; e e ). Again by

54 Thangavelu P. and Evanzalin Ebenanjar P. using the same Lemma 2.11, G = ( e 0 F ) 0(e) for some F e 2 e. That is G = F e (e) is soft closed in (X; E; e) for some soft closed set F e in (X; E; e). Proposition 3.3 Let (X; E; e) be a soft topological space. Let F e 2 S(X; E). Then escl F e (e) = cl(f (e)) in (X; ee ) for every e 2 E. \ (escl F e )(e) = fh(e) e : H e F e ; H e is soft closedg \ = f H(e) e : H(e) e F e (e); H e is soft closedg \ = f H(e) e : H(e) e F e (e); H(e) e is closed in ee g = cl( F e (e)): This shows that escl F e (e) = cl(f (e)) in (X; ee ) for every e 2 E. Proposition 3.4 Let (X; E; e) be a soft topological space. Let F e 2 S(X; E). Then esint F e (e) = int(f (e)) in (X; ee ) for every e 2 E. [ (esint F e )(e) = fg(e) e : G e F e ; G e is soft openg [ = f G(e) e : G(e) e F e (e); G e is soft openg [ = f G(e) e : G(e) e F e (e); G(e) e is open in ee g = int( F e (e)): Therefore esint F e (e) = int(f (e)) in (X; ee ) for every e 2 E. Proposition 3.5 Let (X; E; e) be a soft topological space over X. Let e F 2 S(X; E). Then e F is soft closed in (X; E; e) if and only if e F (e) is closed in (X; e e ).

On topologies induced by the soft topology 55 Suppose e F 2 S(X; E) is soft closed in (X; E; e), escl e F = e F, (escl e F )(e) = ef (e) for every e 2 E, cl( e F (e)) = e F (e) for every e 2 E by using Proposition 3.3., e F (e) is closed in (X; ee ). Proposition 3.6 Let (X; E; e) be a soft topological space over X. Let e F 2 S(X; E). Then e F is soft open in (X; E; e) if and only if e F (e) is open in (X; ee ). Suppose e F 2 S(X; E) is soft open in (X; E; e), esint e F = e F, (esint e F )(e) = ef (e) for every e 2 E, int( e F (e)) = e F (e) for every e 2 E by using Proposition 3.4, e F (e) is open in (X; ee ). Denition 3.7 A family f e F : 2 g of soft sets over X is soft locally nite in the soft topological space (X; E; e) if for every x 2 X there exists a soft open set e G such that x 2 e G and e G \ e F 6= ë holds for at most nitely many indices. The next proposition gives the connection between soft locally nite family in soft topological spaces and locally nite family in topological spaces induced by the soft topology. Proposition 3.8 Let (X; E; e) be a soft topological space over X. Let f e F : 2 g be a family of soft sets over X. Then f e F : 2 g is soft locally nite in (X; E; e) i ff (e) : 2 g is locally nite in the topological space (X; e e ) for every e 2 E. Suppose f e F : 2 g is soft locally nite in (X; E; e)., for every x 2 X there exists a soft open set e G such that x 2 e G and e G \ e F 6= ë holds for at most nitely many indices., x 2 e G(e) and e G(e) \ e F (e) 6= for atmost nitely many indices, for every e 2 E, f e F (e) : 2 g is locally nite in (X; e e ) for every e 2 E.

56 Thangavelu P. and Evanzalin Ebenanjar P. Theorem 3.9 Let f e F ; 2 g be a soft locally nite in the soft topological space (X; E; e). Then fescl( e F ); 2 g is also soft locally nite. Suppose f F e : 2 g is soft locally nite. By using Proposition 3.8, f F e (e) : 2 g is locally nite in (X; e e ). Then fcl( F e (e)) : 2 g is locally nite in (X; e e ) for every e 2 E. By using Theorem 9.2(1) of [5] page 82, f(escl F e )(e) : 2 g is locally nite in (X; e e ) for every e 2 E, that implies fescl F e : 2 g is soft locally nite in (X; E; e). Theorem 3.10 Let f e Fi ; i 2 g be soft locally nite in the soft topological space (X; E; e). Then [escl( e Fi ) is soft closed set. Suppose f e F : 2 g is soft locally nite. By using the Proposition 3.8, f e F (e) : 2 g is locally nite in (X; e e ) for every e 2 E. Then by using Theorem 9.2(2) of [5] page 82, [fcl( e F (e)) : 2 g is closed in (X; e e ) for every e 2 E. By using Proposition 3.3, [f(escl e F )(e) : 2 g is closed in (X; e e ) for every e 2 E. That is ([fescl e F : 2 g)(e) is closed in (X; e e ) for every e 2 E. Then by using Proposition 3.5, [fescl e F : 2 g is soft closed in (X; E; e). Theorem 3.11 Let f e F ; 2 g be a soft locally nite in the soft topological space (X; E; e). Then [fescl( e F ) : 2 g = escl([ e F : 2 ). Suppose f e F : 2 g is soft locally nite. By using Proposition 3.8, f e F (e) : 2 g is locally nite in (X; e e ) for every e 2 E. By using Theorem 3.10 and by using Theorem 9.2(2) of [5] page 82, [fcl( e F (e)) : 2 g is closed in (X; e e ) for every e 2 E. Again by using Problem 7 of [5] page 91, [fcl( e F )(e) : 2 g = clf[( e F )(e) : 2 g in (X; e e ) for every e 2 E. By using Proposition 3.3, ([(escl e F ) : 2 g)(e) = (escl [ f e F : 2 g)(e) in (X; e e ) for every e 2 E. That is ([fescl e F g)(e) = (escl [ f e F g)(e) in (X; e e ) for every e 2 E. Therefore [fescl e F : 2 g = escl [ f e F : 2 g in (X; E; e).

On topologies induced by the soft topology 57 Lemma 3.12 Let (X; E; e) be a soft topological spaces. Let Z X. Then (Z; E; e Z ) is a soft subspace of (X; E; e). Then (e Z ) e = e e Z. B 2 (e Z ) e, B = e F (e) for some e F 2 e Z, B = e H(e) \ e Z(e) where ef = e H \ e Z, e H 2 e, B = H(e) \ Z where H(e) 2 ee, B 2 e e Z : Theorem 3.13 Let fx ; 2 g be a family of sets that cover the set X. That is X = [X. Let (X; E; e) be a soft topological space. Assume that all the ex are soft open sets and let B X. Then e B is soft open if and only if each eb \ e X is soft open in the soft subspace (X ; E; e X ). Suppose X e is soft open in (X; E; e) for every 2. Then by using Proposition 3.6, X is open in (X; e e ) for every e 2 E. That is fx : 2 g is an open cover for (X; e e ) for every e 2 E. By using Theorem 9.3 of [5] page 82, B Xis open in (X; e e ) if and only if each B \ X is open in the subspace (X ; e e ) for every e 2 E. Since by using Lemma 3.12, we have X B X is open in (X; (e) e ) if and only if each B \ X is open in the subspace (X ; (e) e ) for every X e 2 E. That is B e is soft open in (X; E; e) if and only if e B \ e X is soft open in the subspace (X ; E; e X ). Theorem 3.14 Let fx ; 2 g be a family of sets that cover the set X. That is X = [X. Let (X; E; e) be a soft topological space. Assume that all the e X are soft closed sets and form a soft locally nite family and let B X. Then eb is soft closed if and only if each e B \ e X is soft closed in the soft subspace (X ; E; e X ). Suppose X e is soft closed in (X; E; e) for every 2. Then by using Proposition 3.5, X is closed in (X; e e ) for every e 2 E. Then fx : 2 g is locally nite and X is closed in (X; e e ) for every e 2 E. Then using Theorem 9.3 of [5] page 82,B X isclosed in (X; e e ) if and only if each B \ X is closed in the subspace (X ; e e ) for every e 2 E. Since by Lemma 3.12, X

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