aa r X i v : . [ m a t h . GN ] A p r AN EMBEDDING LEMMA IN SOFT TOPOLOGICAL SPACES
GIORGIO NORDO
Abstract.
In 1999, Molodtsov initiated the concept of Soft Sets Theory as anew mathematical tool and a completely different approach for dealing withuncertainties in many fields of applied sciences. In 2011, Shabir and Nazintroduced and studied the theory of soft topological spaces, also definingand investigating many new soft properties as generalization of the classicalones. In this paper, we introduce the notions of soft separation between softpoints and soft closed sets in order to obtain a generalization of the well-knownEmbedding Lemma for soft topological spaces. Introduction
Many practical problems in economics, engineering, environment, social science,medical science etc. cannot be studied by classical methods, because they haveinherent difficulties due to the inadequacy of the theories of parameterization toolsin dealing with uncertainties. In 1999, Molodtsov [14] initiated the novel concept ofSoft Sets Theory as a new mathematical tool and a completely different approachfor dealing with uncertainties while modelling problems in computer science, engi-neering physics, economics, social sciences and medical sciences. Molodtsov definesa soft set as a parameterized family of subsets of universe set where each elementis considered as a set of approximate elements of the soft set.In 2011, Shabir and Naz [16] introduced the concept of soft topological spaces,also defining and investigating the notions of soft closed sets, soft closure, softneighborhood, soft subspace and some separation axioms. Some other propertiesrelated to soft topology were studied by C¸ a˘gman, Karata¸s and Enginoglu in [3].In the same year Hussain and Ahmad [8] continued the study investigating theproperties of soft closed sets, soft neighbourhoods, sof interior, soft exterior andsoft boundary.In the present paper we will present the notions of family of soft mappingssoft separating soft points and soft points from soft closed sets in order to give ageneralization of the well-known Embedding Lemma for soft topological spaces.2.
Preliminaries
In this section we present some basic definitions and results of the theories ofsoft sets and soft topological spaces, simplifying them in a suitable way wheneverpossible. Terms and undefined concepts are used as in [6].
Definition 2.1. [14]
Let U be an initial universe set and E be a nonempty setof parameters (or abstract attributes) under consideration with respect to U and A ⊆ E , we say that a pair ( F, A ) is a soft set over U if F is a set-valued mapping F : A → P ( U ) which maps every parameter e ∈ A to a subset F ( e ) of U . In other words, a soft set is not a real (crisp) set but a parameterized family { F ( e ) } e ∈ A of subsets of the universe U . For every parameter e ∈ A , F ( e ) may beconsidered as the set of e -approximate elements of the soft set ( F, A ). Remark 2.1.
In 2010, Ma, Yang and Hu [11] proved that every soft set ( F, A ) is equivalent to the soft set ( F, E ) related to the whole set of parameters E , simplyconsidering empty every approximations of parameters which are missing in A , thatis extending in a trivial way its set-valued mapping, i.e. setting F ( e ) = ∅ , for every e ∈ E \ A .For such a reason, in this paper we can consider all the soft sets over the sameparameter set E as in [4] and we will redefine all the basic operations and relationsbetween soft sets originally introduced in [14, 12, 13] as in [15] , that is by consideringthe same parameter set. Definition 2.2. [18]
The set of all the soft sets over a universe U with respect toa set of parameters E will be denoted by SS ( U ) E . Definition 2.3. [15]
Let ( F, E ) , ( G, E ) ∈ SS ( U ) E be two soft sets over a commonuniverse U and a common set of parameters E , we say that ( F, E ) is a soft subset of ( G, E ) and we write ( F, E ) ˜ ⊆ ( G, E ) if F ( e ) ⊆ G ( e ) for every e ∈ E . Definition 2.4. [15]
Let ( F, E ) , ( G, E ) ∈ SS ( U ) E be two soft sets over a com-mon universe U , we say that ( F, E ) and ( G, E ) are soft equal and we write ( F, E ) ˜=( G, E ) if ( F, E ) ˜ ⊆ ( G, E ) and ( G, E ) ˜ ⊆ ( F, E ) . Definition 2.5. [15]
A soft set ( F, E ) over a universe U is said to be null softset and it is denoted by (˜ ∅ , E ) if F ( e ) = ∅ for every e ∈ E . Definition 2.6. [15]
A soft set ( F, E ) ∈ SS ( U ) E over a universe U is said to be a absolute soft set and it is denoted by ( ˜ U , E ) if F ( e ) = U for every e ∈ E . Definition 2.7. [15]
Let ( F, E ) ∈ SS ( U ) E be a soft set over a universe U , the softcomplement (or more exactly the soft relative complement) of ( F, E ) , denoted by ( F, E ) ∁ , is the soft set (cid:16) F ∁ , E (cid:17) where F ∁ : E → P ( U ) is the set-valued mappingdefined by F ∁ ( e ) = F ( e ) ∁ = U \ F ( e ) , for every e ∈ E . Definition 2.8. [15]
Let ( F, E ) , ( G, E ) ∈ SS ( U ) E be two soft sets over a commonuniverse U , the soft difference of ( F, E ) and ( G, E ) , denoted by ( F, E ) e \ ( G, E ) , isthe soft set ( F \ G, E ) where F \ G : E → P ( U ) is the set-valued mapping definedby ( F \ G )( e ) = F ( e ) \ G ( e ) , for every e ∈ E . Clearly, for every soft set ( F, E ) ∈ SS ( U ) E , it results ( F, E ) ∁ ˜= ( ˜ U , E ) e \ ( F, E ). Definition 2.9. [15]
Let { ( F i , E ) } i ∈ I ⊆ SS ( U ) E be a nonempty subfamily of softsets over a universe U , the (generalized) soft union of { ( F i , E ) } i ∈ I , denoted by eS i ∈ I ( F i , E ) , is defined by (cid:0)S i ∈ I F i , E (cid:1) where S i ∈ I F i : E → P ( U ) is the set-valuedmapping defined by (cid:0)S i ∈ I F i (cid:1) ( e ) = S i ∈ I F i ( e ) , for every e ∈ E . Definition 2.10. [15]
Let { ( F i , E ) } i ∈ I ⊆ SS ( U ) E be a nonempty subfamily ofsoft sets over a universe U , the (generalized) soft intersection of { ( F i , E ) } i ∈ I ,denoted by eT i ∈ I ( F i , E ) , is defined by (cid:0)T i ∈ I F i , E (cid:1) where T i ∈ I F i : E → P ( U ) is theset-valued mapping defined by (cid:0)T i ∈ I F i (cid:1) ( e ) = T i ∈ I F i ( e ) , for every e ∈ E . Definition 2.11. [9]
Two soft sets ( F, E ) and ( G, E ) over a common universe U are said to be soft disjoint if their soft intersection is the soft null set, i.e. if ( F, E )˜ ∩ ( G, E ) ˜= (˜ ∅ , E ) . Definition 2.12. [17]
A soft set ( F, E ) ∈ SS ( U ) E over a universe U is said tobe a soft point over U if it has only one non-empty approximation and it is asingleton, i.e. if there exists some parameter α ∈ E and an element p ∈ U suchthat F ( α ) = { p } and F ( e ) = ∅ for every e ∈ E \ { α } . Such a soft point is usually N EMBEDDING LEMMA IN SOFT TOPOLOGICAL SPACES 3 denoted by ( p α , E ) . The singleton { p } is called the support set of the soft point and α is called the expressive parameter of ( p α , E ) . Definition 2.13. [17]
The set of all the soft points over a universe U with respectto a set of parameters E will be denoted by SP ( U ) E . Since any soft point is a particular soft set, it is evident that SP ( U ) E ⊆ SS ( U ) E . Definition 2.14. [17]
Let ( p α , E ) ∈ SP ( U ) E and ( F, E ) ∈ SS ( U ) E respectively bea soft point and a softset over a common universe U . We say that the soft point ( p α , E ) soft belongs to the soft set ( F, E ) and we write ( p α , E )˜ ∈ ( F, E ) , if the softpoint is a soft subset of the soft set, i.e. if ( p α , E ) ˜ ⊆ ( F, E ) and hence if p ∈ F ( α ) . Definition 2.15. [5]
Let ( p α , E ) , ( q β , E ) ∈ SP ( U ) E be two soft points over a com-mon universe U , we say that ( p α , E ) and ( q β , E ) are soft equal , and we write ( p α , E ) ˜=( q β , E ) , if they are equals as soft sets and hence if p = q and α = β . Definition 2.16. [5]
We say that two soft points ( p α , E ) and ( q β , E ) are softdistincts , and we write ( p α , E ) ˜ =( q β , E ) , if and only if p = q or α = β . According to Remark 2.1 the following definitions by Kharal and Ahmad havebeen simplified and slightly modified for soft sets on a common parameter set.
Definition 2.17. [10]
Let SS ( U ) E and SS ( U ′ ) E ′ be two sets of soft open sets overthe universe sets U and U ′ with respect to the sets of parameters E and E ′ , re-spectively. and consider a mapping ϕ : U → U ′ between the two universe setsand a mapping ψ : E → E ′ between the two set of parameters. The mapping ϕ ψ : SS ( U ) E → SS ( U ′ ) E ′ which maps every soft set ( F, E ) of SS ( U ) E to a soft set ϕ ψ (( F, E )) of SS ( U ′ ) E ′ denoted by ( ϕ ψ ( F ) , E ′ ) where ϕ ψ ( F ) : E ′ → P ( U ′ ) is theset-valued mapping defined by ϕ ψ ( F )( e ′ ) = S e ∈ ψ − ( { e ′ } ) ϕ ( F ( e )) for every e ′ ∈ E ′ ,is called a soft mapping from U to U ′ induced by the mappings ϕ and ψ , while thesoft set ϕ ψ ( F, E ) ˜=( ϕ ψ ( F ) , E ′ ) is said to be the soft image of the soft set ( F, E ) under the soft mapping ϕ ψ : SS ( U ) E → SS ( U ′ ) E ′ .The soft mapping ϕ ψ : SS ( U ) E → SS ( U ′ ) E ′ is said injective (respectively sur-jective , bijective ) if the mappings ϕ : U → U ′ and ψ : E → E ′ are both injective(resp. surjective, bijective). Definition 2.18. [10]
Let ϕ ψ : SS ( U ) E → SS ( U ′ ) E ′ be a soft mapping induced bythe mappings ϕ : U → U ′ and ψ : E → E ′ between the two sets SS ( U ) E , SS ( U ′ ) E ′ ofsoft sets and consider a soft set ( G, E ′ ) of SS ( U ′ ) E ′ . The soft inverse image of ( G, E ′ ) under the soft mapping ϕ ψ : SS ( U ) E → SS ( U ′ ) E ′ , denoted by ϕ − ψ (( G, E ′ )) is the soft set ( ϕ − ψ ( G ) , E ′ ) of SS ( U ) E where ϕ − ψ ( G ) : E → P ( U ) is the set-valuedmapping defined by ϕ − ψ ( G )( e ) = ϕ − ( G ( ψ ( e ))) for every e ∈ E . The concept of soft topological space as topological space defined by a familyof soft sets over a initial universe with a fixed set of parameters was introduced in2011 by Shabir and Naz [16].
Definition 2.19. [16]
Let X be an initial universe set, E be a nonempty set ofparameters with respect to X and T ⊆ SS ( X ) E be a family of soft sets over X ,we say that T is a soft topology on X with respect to E if the following fourconditions are satisfied:(i) the null soft set belongs to T , i.e. (˜ ∅ , E ) ∈ T (ii) the absolute soft set belongs to T , i.e. ( ˜ X, E ) ∈ T (iii) the soft intersection of any two soft sets of T belongs to T , i.e. for every ( F, E ) , ( G, E ) ∈ T then ( F, E )˜ ∩ ( G, E ) ∈ T . GIORGIO NORDO (iv) the soft union of any subfamily of soft sets in T belongs to T , i.e. for every { ( F i , E ) } i ∈ I ⊆ T then eS i ∈ I ( F i , E ) ∈ T The triplet ( X, T , E ) is called a soft topological space over X with respect to E .In some case, when it is necessary to better specify the universal set and the set ofparameters, the topology will be denoted by T ( X, E ) . Definition 2.20. [16]
Let ( X, T , E ) be a soft topological space over X with respectto E , then the members of T are said to be soft open set in X . Definition 2.21. [7]
Let T and T be two soft topologies over a common universeset X with respect to a set of paramters E . We say that T is finer (or stronger)than T if T ⊆ T where ⊆ is the usual set-theoretic relation of inclusion betweencrisp sets. In the same situation, we also say that T is coarser (or weaker) than T . Definition 2.22. [16]
Let ( X, T , E ) be a soft topological space over X and be ( F, E ) be a soft set over X . We say that ( F, E ) is soft closed set in X if its complement ( F, E ) ∁ is a soft open set, i.e. if ( F ∁ , E ) ∈ T . Notation 2.1.
The family of all soft closed sets of a soft topological space ( X, T , E ) over X with respect to E will be denoted by σ , or more precisely with σ ( X, E ) whenit is necessary to specify the universal set X and the set of parameters E . Definition 2.23. [2]
Let ( X, T , E ) be a soft topological space over X and B ⊆ T be a non-empty subset of soft open sets. We say that B is a soft open base for ( X, T , E ) if every soft open set of T can be expressed as soft union of a subfamilyof B , i.e. if for every ( F, E ) ∈ T there exists some A ⊂ B such that ( F, E ) = eT { ( A, E ) : ( A, E ) ∈ A} . Definition 2.24. [18]
Let ( X, T , E ) be a soft topological space, ( N, E ) ∈ SS ( X ) E be a soft set and ( x α , E ) ∈ SP ( X ) E be a soft point over a common universe X . Wesay that ( N, E ) is a soft neighbourhood of the soft point ( x α , E ) if there is somesoft open set soft containing the soft point and soft contained in the soft set, thatis if there exists some soft open set ( A, E ) ∈ T such that ( x α , E )˜ ∈ ( A, E ) ˜ ⊆ ( N, E ) . Definition 2.25. [16]
Let ( X, T , E ) be a soft topological space over X and ( F, E ) be a soft set over X . Then the soft closure of the soft set ( F, E ) , denoted by s - cl X (( F, E )) , is the soft intersection of all soft closed set over X soft containing ( F, E ) , that is s - cl X (( F, E )) ˜= f\ (cid:8) ( C, E ) ∈ σ ( X, E ) : ( F, E ) ˜ ⊆ ( C, E ) (cid:9) Definition 2.26. [18]
Let ϕ ψ : SS ( X ) E → SS ( X ′ ) E ′ be a soft mapping betweentwo soft topological spaces ( X, T , E ) and ( X ′ , T ′ , E ′ ) induced by the mappings ϕ : X → X ′ and ψ : E → E ′ and ( x α , E ) ∈ SP ( X ) E be a soft point over X . Wesay that the soft mapping ϕ ψ is soft continuous at the soft point ( x α , E ) if foreach soft neighbourhood ( G, E ′ ) of ϕ ψ (( x α , E )) in ( X ′ , T ′ , E ′ ) there exists some softneighbourhood ( F, E ) of ( x α , E ) in ( X, T , E ) such that ϕ ψ (( F, E )) ˜ ⊆ ( G, E ′ ) .If ϕ ψ is soft continuous at every soft point ( x α , E ) ∈ SP ( X ) E , then ϕ ψ : SS ( X ) E →SS ( X ′ ) E ′ is called soft continuous on X . Definition 2.27. [2]
Let ( X, T , E ) be a soft topological space over X and S ⊆ T be a non-empty subset of soft open sets. We say that S is a soft open subbase for ( X, T , E ) if the family of all finite soft intersections of members of S forms asoft open base for ( X, T , E ) . N EMBEDDING LEMMA IN SOFT TOPOLOGICAL SPACES 5
Proposition 2.1. [2]
Let
S ⊆ SS ( X ) E be a a family of soft sets over X andcontaining both the null soft set (˜ ∅ , E ) and the absolute soft set ( ˜ X, E ) . Then thefamily T ( S ) of all soft union of finite soft intersections of soft sets in S is a softtopology having S as subbase. Definition 2.28. [2]
Let
S ⊆ SS ( X ) E be a a family of soft sets over X respectto a set of parameters E and such that (˜ ∅ , E ) , ( ˜ X, E ) ∈ S , then the soft topology T ( S ) of the above Proposition 2.1 is called the soft topology generated by thesoft subbase S over X and ( X, T ( S ) , E ) is said to be the soft topological spacegenerated by S . An Embedding Lemma for soft topological spaces
Definition 3.1. [2]
Let SS ( X ) E be the set of soft sets over a universe set X withrespect to a set of parameter E and consider a family of soft topological spaces { ( Y i , T i , E i ) } i ∈ I and a corresponding family { ( ϕ ψ ) i } i ∈ I of soft mappings ( ϕ ψ ) i : SS ( X ) E → SS ( Y i ) E i . Then the soft topology T ( S ) generated by the soft subbase S = (cid:8) ( ϕ ψ ) − i (( G, E i )) : ( G, E i )˜ ∈T i , i ∈ I (cid:9) of all soft inverse images of the softmappings ( ϕ ψ ) i is called the initial soft topology induced on X by the family ofsoft mappings { ( ϕ ψ ) i } i ∈ I and it is denoted by T ini ( X, E , Y i , E i , ( ϕ ψ ) i ; i ∈ I ) . Definition 3.2.
Let { ( X i , T i , E i ) } i ∈ I be a family of soft topological spaces over theuniverse sets X i with respect to the sets of parameters E i , respectively. For every i ∈ I , the soft mapping ( π i ) ρ i : SS ( Q i ∈ I X i ) Q i ∈ I E i → SS ( X i ) E i induced by theclassical projections π i : Q i ∈ I X i → X i and ρ i : Q i ∈ I E i → E i is said the i -thsoft projection mapping and, by setting ( π ρ ) i = ( π i ) ρ i , it will be denoted by ( π ρ ) i : SS ( Q i ∈ I X i ) Q i ∈ I E i → SS ( X i ) E i . Definition 3.3. [2]
Let { ( X i , T i , E i ) } i ∈ I be a family of soft topological spaces andconsider the corresponding family { ( π ρ ) i } i ∈ I of soft projection mappings ( π ρ ) i : SS ( Q i ∈ I X i ) Q i ∈ I E i → SS ( X i ) E i (with i ∈ I ). Then, the initial soft topology T ini (cid:0)Q i ∈ I X i , E , X i , E i , ( π ρ ) i ; i ∈ I (cid:1) induced on Q i ∈ I X i by the family of soft pro-jection mappings { ( π ρ ) i } i ∈ I is called the soft topological product of the softtopological space ( X i , T i , E i ) (with i ∈ I ) and denoted by T (cid:0)Q i ∈ I X i (cid:1) . Definition 3.4. [1]
Let ( X, T , E ) and ( X ′ , T ′ , E ′ ) be two soft topological spacesover the universe sets X and X ′ with respect to the sets of parameters E and E ′ , respectively. We say that a soft mapping ϕ ψ : SS ( X ) E → SS ( X ′ ) E ′ is a soft homeomorphism if it is soft continuous, bijective and its inverse ϕ − ψ : SS ( X ′ ) E ′ : → SS ( X ) E is a soft continuous mapping too. In such a case, the softtopological spaces ( X, T , E ) and ( X ′ , T ′ , E ′ ) are said soft homeomorphic and wewrite that ( X, T , E ) ˜ ≈ ( X ′ , T ′ , E ′ ) . Definition 3.5.
Let ( X, T , E ) and ( X ′ , T ′ , E ′ ) be two soft topological spaces. Wesay that a soft mapping ϕ ψ : SS ( X ) E → SS ( X ′ ) E ′ is a soft embedding if itscorestriction ϕ ψ : SS ( X ) E → ϕ ψ ( SS ( X ) E ) is a soft homeomorphism. Definition 3.6.
Let ( X, T , E ) be a soft topological space over a universe set X withrespect to a set of parameter E , let { ( X i , T i , E i ) } i ∈ I be a family of soft topologicalspaces over a universe set X i with respect to a set of parameters E i , respectivelyand consider a family { ( ϕ ψ ) i } i ∈ I of soft mappings ( ϕ ψ ) i = ( ϕ i ) ψ i : SS ( X ) E →SS ( X i ) E i induced by the mappings ϕ i : X → X i and ψ i : E → E i (with i ∈ I ).Then the soft mapping ∆ = ϕ ψ : SS ( X ) E → SS ( Q i ∈ I X i ) Q i ∈ I E i induced by thediagonal mappings (in the classical meaning) ϕ = ∆ i ∈ I ϕ i : X → Q i ∈ I X i and GIORGIO NORDO ψ = ∆ i ∈ I ψ i : E → Q i ∈ I E i (respectively defined by ϕ ( x ) = h ϕ i ( x ) i i ∈ I for every x ∈ X and by ψ ( e ) = h ψ i ( e ) i i ∈ I for every x ∈ X ) is called the soft diagonalmapping of the soft mappings ( ϕ ψ ) i (with i ∈ I ) and denoted by ∆ = ∆ i ∈ I ( ϕ ψ ) i : SS ( X ) E → SS ( Q i ∈ I X i ) Q i ∈ I E i . Definition 3.7.
Let { ( ϕ ψ ) i } i ∈ I be a family of of soft mappings ( ϕ ψ ) i : SS ( X ) E →SS ( X i ) E i between a soft topological space ( X, T , E ) and a family of soft topo-logical spaces { ( X i , T i , E i ) } i ∈ I . We say that the family { ( ϕ ψ ) i } i ∈ I soft sepa-rates soft points of ( X, T , E ) if for every ( x α , E ) , ( y β , E )( ∈ SP ( X ) E such that ( x α , E ) ˜ =( y α , E ) there exists some i ∈ I such that ( ϕ ψ ) i ( x α , E ) ˜ =( ϕ ψ ) i ( y β , E ) . Definition 3.8.
Let { ( ϕ ψ ) i } i ∈ I be a family of of soft mappings ( ϕ ψ ) i : SS ( X ) E →SS ( X i ) E i between a soft topological space ( X, T , E ) and a family of soft topologicalspaces { ( X i , T i , E i ) } i ∈ I . We say that the family { ( ϕ ψ ) i } i ∈ I soft separates softpoints from soft closed sets of ( X, T , E ) if for every ( C, E ) ∈ σ ( X, E ) andevery ( x α , E )( ∈ SP ( X ) E such that ( x α , E )˜ ∈ ( ˜ X, E ) e \ ( C, E ) there exists some i ∈ I such that ( ϕ ψ ) i ( x α , E )˜ / ∈ s - cl X i (( ϕ ψ ) i ( C, E )) . Proposition 3.1 ( Soft Embedding Lemma ) . Let ( X, T , E ) be a soft topologicalspace, { ( X i , T i , E i ) } i ∈ I be a family of soft topological spaces and { ( ϕ ψ ) i } i ∈ I be afamily of of soft continuous mappings ( ϕ ψ ) i : SS ( X ) E → SS ( X i ) E i that separatesboth the soft points and the soft points from the soft closed sets of ( X, T , E ) . Thenthe diagonal mapping ∆ = ∆ i ∈ I ( ϕ ψ ) i : SS ( X ) E → SS ( Q i ∈ I X i ) Q i ∈ I E i of the softmappings ( ϕ ψ ) i is a soft embedding. Conclusion
In this short announcement paper we have introduced the notions of family ofsoft mappings separating points and points from closed sets and that of soft diagonalmapping in order to define the necessary framework for proving a generalization tosoft topological spaces of the well-known Embedding Lemma for classical (crisp)topological spaces. Such a result could be the start point for investigating otherimportant topics in soft topology such as extension and compactifications theorems,metrization theorems etc. Details and proofs will be given in a next extended paper.
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