On Soft Semi Weakly Generalized Closed Set in Soft TopologicalSpaces

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International Journal of Innovative Technology and Exploring Engineering (IJITEE) ISSN: 2278-3075, Volume-8 Issue-12, October, 2019

Abstract—Soft sets has helped the development of soft topological space and it was also applied in the ﬁeld of life science, Social science and Engineering. Many researchers developed various ideas based on the properties of soft topology. The article deals with study of properties in soft topological space based on soft semi weakly generalized closed set.

Keywords—Soft closed set, Softopen set, Soft swg-closure set, Soft swg-interior.

Subject Classification:06D72,54A05

I. INTRODUCTION

Intheyear1999,the idea of soft set theory was introduced by Molodstov[2]tofindanswerstoseveral problemsinlifescience,engineering,and in practical life situationetc.,Later theresults wereapplied indifferent fieldsoft study viz., operations research, gametheoryetc.,. Ça˘gmanN[1]introduced softtopologyfromwhich manyresearchers applieditas abase toworkon softtopologicalspaceanditwasthe

beginningforsoftmathematicalconcepts.Here we introduced the idea of soft topological space in the basis of soft semi weakly generalized closed set.Thecollectionof soft sets overFA was stuided by Shabir M and Naz M, [7] and denotedfewnotionsofttopologicalspace.

Thepropertiesofsofttopologicalspaces werestudiedbythe authors [3-8]

LetU be an universal set and E set of parameters; P(U)the power set of U. The collection of all soft sets over Uand Eis denoted by S U . If A E, then the pair(F, A)is said to be the soft set over U and it is denoted byFAor FE where F is a mapping of A ontoP(U). Note that for eA,F e = F∅ [2].Let FB and GC be the soft sets in a universe set Uand B, C ⊆E. Soft subset of GC represented 𝐹𝐵, symbolizedby F B GC, when (i) B⊆C and (ii) ∀ eB, F(e)=G(e).The relative complement of a soft set FA, denoted by FAc, is being represented by thefunction fAc e = fAc e , that is fAc e = U − fA e ∀ eE. In other words(FAc)c= FA,Fϕc = FEand FEc = Fϕ[1]. Let FA∈ S(U). A soft topology on (FA, τ ), which is a group of soft subsets of FAhas the following properties(i)

Fϕ , FA ∈ τ .(ii) FA_i FA∶ i ∈ I  τ Ũi∈IFA_i ∈ τ . (iii) FA_i FA∶ 1 ≤ i ≤ n , n ∈   τ ∩i=1n FA_i ∈ τ .The pair (FA, τ ) is known assoft topological spaces [7].Let(FA,

Revised Manuscript Received on August 05, 2019.

Sasikala V.E, Research Scholar, Vels Instituteof Science, Technology and Advanced Studies, Chennai, Tamil nadu, India.

(E-mail: [email protected])

Sivaraj .D, Meenakshi Academy of Higher Education and Research, Chennai ,Tamil Nadu, India.

Ponraj A.P, Meenakshi Academy of Higher Education and Research, Chennai ,Tamil Nadu, India.

τ )be a soft topological space and FB be a soft set overFA. (i) Soft interior of FB is the soft set int(FB) = Ũ{FC: FCis soft open set and FB FC}. (ii) Soft closure of FBis the soft set cl(FB)=∩ {FC: FCis soft closed set and FC FB} [7]. (i) Soft α-closed set, when cl(int(cl(FB))) FB [3]. (ii) Soft β −closed set (or) Soft semi-pre closed set, when int(cl(int(FB))) FB [3]. (iii) Soft semi-closed set, when int(cl(FB)) FB [5]. (iv) Soft pre-closed set, when cl(int(FB)) FB [6]. (v) Soft regular closed set, when cl(int(FB))= FB [8]. (vi) Soft generalized closed set, when cl(FB) FC, whenever FB FCand FCis a soft open set and Soft weakly closed set, if every cl(FB) FC, whenever FB FC and FCis a soft semi-open set [8].

II. SOFT SEMI WEAKLY GENERALIZED CLOSED SET AND SOFT SEMI

WEAKLYGENERALIZED CLOSED SETS IN THEIR PROPERTIES

Definition 2.1.Let (FA , τ ) is a soft topological space and FB FA,FB is identified to be a Soft Semi Weakly Generalized closed set(brieﬂy,soft swg-closed set) if every cl(int(FB)) FC, whenever FB FCand FC (soft semi-open set).

Example 2.2.Let U = {𝑎,b,c}, E = {e1, e2, e3}, A = {e1, e2 }  E.

FA = {(e1, {𝑎,b,c}), (e2, {𝑎,b,c})},

F1 = {(e1, {b,c}), (e2, {b,c})}, F2 = {(e1, {c}), (e2, {𝑎,b})}, F3 = {(e1, {𝑎,c}), (e2, {b})}, F4= {(e1, {c}), (e2, {b})},F5= {(e1, {b,c}), (e2, {𝑎,b,c})},F6={(e1, {𝑎,b,c}), (e2, {b,c})},F7 = {(e1, {𝑎,c}), (e2, {𝑎,b})}, F8= {(e1, {𝑎}), (e2, {𝑎})}, F9 = {(e1, {𝑎,b}), (e2, {c})}, F10={(e1,{b}), (e2, {𝑎,c})},F11={(e1, {𝑎,b}), (e2, {𝑎,c})},F12= {(e1, {𝑎})}, F13= {(e2, {𝑎})}, F14={(e1, {b}), (e2, {c})},F15 =FA, F16= Fϕ. Let τ = {Fϕ, FA, F1,F2,F3, F4, F5,F6, F7}. Then (FA, τ )is a soft topological space. Soft open set are {FA, Fϕ, F1,F2,F3, F4, F5,F6, F7}. Soft closed set are {FA, Fϕ, F8,F9,F10, F11, F12,F13, F14}.

(i) Let us consider FB = F8= {(e1, {𝑎}), (e2, {𝑎})},F8 FAand FC= {(e1, {𝑎}), (e2, {𝑎,𝑏})}whereFCis soft semi-open set, thenint(FB) = Fϕ, cl(int(FB)) = Fϕ. So cl(int(FB)) FC.FBissoft semi weakly generalized closedset (soft swg-closed set) but not soft semi-open set.

(ii) Let us considerFB = {(e1, {c}), (e2, {b, c})} and FC= {(e1, {a, b, c}), (e2, {b, c})}whereFC is a soft semi-open set,thenint(FB) = F4, cl(int(FB)) = FA. So,cl(int(FB)) ⊈ FC. FBis a soft semi-open set but it is neither a soft semiweakly generalized closed set(soft swg-

On Soft Semi Weakly Generalized Closed Set in

Soft TopologicalSpaces

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closed set) nor soft closed set orsoft open set.

Theorem 2.3.All soft closed sets represent soft swg-closed set.

Proof.LetFBbe a soft closed set in(FA, τ ). Now int(FB) FBandFB FCalways, if FBis a soft closed set and soft open set. So, if FB FC, where FC is a soft semi-open set in FA, then cl(int(FB))cl(FB) FC. Thus cl(int(FB)) FC, whenever FB FC, whereFC denotes soft semi-open set. Therefore, FBis a soft swg-closed set.□

Example 2.4.We have following example 2.2., LetFB= {(e1, {c}), (e2, {𝑎, b, c})} and FC= {(e1, {𝑎, c}), (e2, {𝑎, b, c})}, where FCis a soft semi-open set, then int(FB)= Fϕ,cl(int(FB)) = Fϕ, so, cl(int(FB)) FC. NowFB is soft swg-closed set but it is not a soft swg-closed set.

Theorem 2.5.All soft g-closed sets represent soft swg-closed set.

Proof.Let FBstands soft g-closed set, then cl(FB) FCwhenever FB FCandFCremains a soft open set. Each soft open set denotes soft semi-openset. int(FB) FB, if FBis a soft open set. Now cl(int(FB)) cl(FB) FC. Therefore, FBrefers asasoft swg-closedset.□

Example 2.6.We have following example 2.2., LetFB= {(e1, {c}),(e2, {𝑎})},FC= {(e1, {c}), (e2, {𝑎, b})} and FB FC where FCrepresents soft open set and soft semi-open set.Thenint(FB)= Fϕ,cl(int(FB)) = Fϕ, cl(int(FB)) FC. Therefore,FB be a soft swg-closed set but it is not a soft g-closed set.

Theorem 2.7.All soft regular closed set represent soft swg-closed set.

Proof.Let FBrefer to a soft regular closed set, if cl(int(FB) = FB. SinceFB FC, where FCremains soft semi-openset. Now cl(int(FB)) = FB FC.Therefore, FBrefers as a soft swg-closed set.

Example 2.8.We have following example 2.2., LetFB= {(e1, {𝑎}), (e2, {b})},FC = {(e1, {𝑎}), (e2, {b, c})} and FB FC where FCis soft semi-open set, then int(FB)= Fϕ,cl(int(FB)) = Fϕ,cl(int(FB)) FC. Therefore,FB refers as asoft swg-closed set but it is not a soft regular closed set

Theorem 2.9.The intersection of two soft swg-closed sets gives another soft swg-closed set.

Proof.LetFB and FCstay any two soft swg-closed sets and FD remains any soft semi-open set compriseFBand FC. By deﬁnition of soft swg-closed set,cl(int(FB) FD and cl(int(FC)) FD. Hence cl(int(FB⋂ FC)) cl(int(FB))⋂cl(int(FC)) FD. Therefore, cl(int(FB⋂ FC)) FD. Therefore, FB⋂ FCrefers assoft swg-closed set inFA.□

Remark2.10.If FB and FC are soft swg-closed sets the FB∪ FC does not need be a soft swg-closed set.

Example 2.11.From 2.2., LetFB= {(e1, {𝑐})} andFC = {(e2, {b})}, then FB and FCare soft swg-closed sets.FB∪ FC ={(e1, {𝑐}), (e2, {b})} is not a soft swg-closed set

Theorem 2.12.Let (FA , τ ) be a soft topological space. Then a soft subset FB ofFA is soft swg-closed set in (FA , τ ) iffcl(int(FB)) − FB contains empty soft semi-closed set

Proof.LetFC be a non empty soft semi-closed set subset ofcl(int(FB)) − FB. NowFCcl(int(FB)) − FB. Which implies thatFC cl(int(FB)) ∩ FBc. Since cl(int(FB)) − F =cl(int(F )) ∩ Fc_. _Thus _F __cl(int(F _)). _Now

FC FBcwhich implies that FB FCc.WhenFCcrepresentssoft semi-open set and FBrepresents soft swg-closed set,we have cl(int(FB)) FCc which implies that (cl(int(FB)))c (FCc)𝑐 which implies that FC (cl(int(FB)))cwhich implies that FC cl(int(FB)) ∩ (cl(int(FB)))c = Fϕ which is a contradiction. Therefore, FC= Fϕ.Therefore, cl(int(FB)) − FB contains empty soft semi-closed set. Conversely, Assume cl(int(FB)) − FB containsempty soft semi-closed sets. Let FB FC, where FC is a soft semi-open set. Presume,cl(int(FB)) is not contained in FC. Then cl(int(FB)) ⊈ FC, then cl(int(FB)) ∩ FCc is a non empty soft semi-closed set of cl(int(FB)) − FB, which is a contradiction. Then cl(int(FB))  FC whenever FB FC. Therefore, FBis a soft swg-closed set. □

Theorem 2.13.Thesoft swg-closed sets be a soft regular closed set, ⟺cl(int(FB)) − FB softsemi-closed set.

Proof.Presume that FBbe soft regular closed set. Since cl(int(FB)) = FB. Since cl(int(FB)) − FB = Fϕ besoft regular closed set and it is soft semi-closed set. Conversely, Assume cl(int(FB)) − FBis a soft semi-closed set [From Theorem 2.12]. Since, cl(int(FB)) − FB = Fϕcontains empty soft semi-closed set. Thus FB proved as soft regular closed set.□

Theorem 2.14.Soft topological space(FA, τ ) and FC FB FA.IfFC is a soft swg-closed set in FA, then FC is a soft swg-closed set in relation toFBand mutually soft open set and soft swg-closed are subset of FA, then FC, softswg-closed set relative to FA

Proof. AssumeFC FD and FDbe the soft open set overFA. From the known resultFC FB FA.Then FC FB andFC FDwhich implies thatFC FB⋂ FD. Since FC denoted as a soft swg-closed set relative to FB, cl(int(FC) FB⋂ FD which implies that FB⋂cl(int(FC)) FB⋂ FDwhich implies that FB⋂(cl(int(FC))) FD. ThusFB⋂cl(int(FC))∪ (cl(int(FC)))c FD∪ (cl(int(FC)))cwhic h implies that FB ∪(cl(int(FC)))c FD∪ (cl(int(FC)))c. Since FB is a soft swg-closed set in FA.We have cl(int(FB) FD∪(cl(int(FC)))c. Also, FC FBwhich implies that cl(int(FC))cl(int(FB)). Thus cl(int(FC)) cl(int(FB)) FD∪ (cl(int(FC)))c. Hence,FC proved as soft swg-closed set in relationto FA

Theorem 2.15.When soft topological space(FA , τ )and FC FB FA. IfFC is soft swg-closed set relative to FBand FB is soft swg-closed set(FA , τ ).Then FC, soft swg-closed set in relation to (FA , τ )

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Theorem 2.16. Let (FA, τ )considered as soft topological space, then a soft subsetFBinsideFA is soft nowhere dense, then FB, soft swg-closed set in FA

Proof. When a soft subset FB of FA is soft nowhere dense, then int(FB) = Fϕ.Let FB FCwhere FCrepresent soft semi-open set,which denotes that cl(int(FB))= cl(Fϕ) = Fϕ FC. Therefore, FBproved as soft swg-closed set in FA.

Theorem 2.17.Let(FA, τ )be a soft topological space.For every soft subset fA∈ FA, either {fA}refers assoft semi-closed set,or {fA}c, soft swg-closed set in FA, τ

Proof.Assume that {fA} does not belongs to soft semi-closed set of (FA , τ ). Then{fA}cis a soft semi-open set and the only soft semi-open set containinginFA itself. Therefore, cl(int({fA}c)) FAand so {fA}c proved as soft swg-closed set in (FA, τ )

Theorem 2.18.If FB state soft semi-open set and soft swg-closed set inFA,both these constitute soft g-closed set

Proof.Let FBbe both soft swg-closed set and softsemi-open

set. Let FB FCwhere FCis soft open set.Then bydeﬁnition of soft swg-closed set, cl(int(FB)) FC. Since FBis soft semi-open set cl(FB)) cl(int(FB)) FC . Which implies thatcl(FB) FC, where FCshows as soft open set. Hence FB proved as softg-closed set.

Theorem 2.19.Let FB be soft swg-closed set and FDis soft closed set,then FB⋂ FDis softswg-closed set

Proof.Let FBbe a soft swg-closed set and FDis soft closed set. Now, show that FB⋂ FDis soft swg-closed set. Let FB⋂

FD FCwhereFCis soft semi-open set. Since FDis soft closed set, FB⋂ FDis soft closed set in FB which implies thatFB⋂ FD FB. Itimplies thatcl(int(FB⋂ FD )) cl (FB⋂ FD) = FB⋂ FD FC.It implies that cl(int(FB⋂ FD))  FC. Therefore, FB⋂ FD proved as softswg-closed set. Theorem 2.20.Let (FA , τ ) be soft topological space. IfFB is

softswg- closed set in (FA, τ ) and FB FCcl(int(FB)), thenFC is soft swg-closed set in (FA , τ ).

Proof.Given that: FC cl(int(FB)), then cl(int(FC))cl(int(FB)).

Which implies that cl(int(FC )) − FC cl(int(FB)) − FB. Since

FB FCandFB issoft swg-closed set in (FA , τ ) [From Theorem 2.12]. Let cl(int(FB))− FB contains empty soft semi-

closed set and cl(int(FC)) − FC contains no non empty soft

semi-closed set. Therefore, FC proved as soft swg-closed set.

III. SOFT SEMI WEAKLY GENERALIZED OPEN SET AND SOFT SEMI WEAKLY GENERALIZED

OPEN SETS IN THEIR PROPERTIES

Definition 3.1.A soft subset FB of(FA , τ ) refers assoft semi weakly generalized open set(brieﬂy, softswg-open set), if its compliment is soft swg-closed set in (FA , τ ).

Theorem 3.2.A soft subsetFB of (FA , τ ) is softregular open set, then it is soft swg-open set.

Proof.LetFBbe a soft regular open set in τ , then FBcis soft regular closed set. It implies FBc = 𝑐𝑙(𝑖𝑛𝑡 FBc .Let FBc FC, since FC is soft semi-open set in τ . It implies that 𝑐𝑙(𝑖𝑛𝑡 FBc  FCwhere FC representsoft semi-open set. Since FBc representsoft swg-closed set. Therefore,FBproved as soft swg-open set.

Theorem 3.3.A soft subset FB of a soft topological space (FA , τ ) is softswg-open set iffFCint(cl(FB)) when FC FB and FCis a soft semi-closed set.

Proof.Presume that FBis a softswg-open setinFA, then FBcrepresent soft swg-closed set. Let FC be a soft semi-closed set in (FA, τ ) contained in FB. Then FCc is soft semi-open set containing FBc. i.e.,FBc FCc. It implies cl(int(FBc)) FCc.Since FBc is soft swg-closed set. ∴ FC 𝑖𝑛𝑡(𝑐𝑙(FB)). Conversely, suppose FC 𝑖𝑛𝑡(𝑐𝑙(FB)) when FC FB and FCis a soft semi-closed set.Then FCc is soft semi-open set containing FBcand FCc (𝑖𝑛𝑡(𝑐𝑙(FB )))c. It follows FCc (𝑐𝑙(𝑖𝑛𝑡 FBc )). FBcis soft swg-closed set and so, FBproved as soft swg-open set.□

Theorem 3.4.If int(cl(FB)) FC FBand FBis soft swg-open set inFA, then FCis softswg-open set

Proof.Letint(cl(FB)) FC FB it

implies(FB)c (FC)c(int(cl(FB)))𝑐which implies that (FB)c (FC)c(cl(int(FBc))) where FBc is soft swg-closed set in FA and also FCc is soft swg-closed set. Therefore, FC proved as softswg-open setin FA.

Theorem 3.5.WhileFB and FC are soft swg-open sets in FA. Let FB, FC FA. If FCis softswg-open set and int(cl(FC)) FB, then FB⋂ FC is soft swg-open set

Proof.Let FB and FCare soft swg-open sets in FAandFBcand FCc aresoft swg-closed sets in FA. Since FBint(cl(FC)) it implies int(cl(FC)) FB⋂ FC FB[By Theorem 3.4], then FB⋂ FCproved as soft swg-open set

Theorem 3.6.A soft set FBis a soft swg-open sets in FA, ⟺FC= FA,whenFCis soft semi-openset and int(cl(FB))∪ FBc FC

Proof.Let FBbe soft swg-open set. FCbesoft semi-open set and int(cl(FB))∪ FBc FC. Which implies that (FC)c (int(cl(FB))∪ FBc)c= (int(cl(FB)))c⋂ FB = (int(cl(FB)))c− FBc=cl(int(FBc)) − FBc. Which implies that FCccl(int(FBc)) − FBc. Since FBc represent soft swg-closed set and FCcrepresent soft semi-closed set, based on thatFCc =Fϕ. Therefore, FC=FA. Conversely, assume thatFDis a soft semi-closed set in FA and FD FB. Then int(cl(FB))∪ FBcint(cl(FB))∪ FDc. Since both int(cl(FB)) and FDc are soft semi-openset, their union int(cl(FB))∪ FDc is also a soft semi-open set. It follows by hypothesis that int(cl(FB))∪ FDc = FA and hence FD int(cl(FB)). Therefore, FBproved as soft swg-open sets in FA

Theorem 3.7.If FB FC FA, where FBis soft swg-open set isin relation to FC and FC is soft swg-open set is in relation to FA, then FB is a softswg-open set is in relation to FA.

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set FEexists such that FD FE⋂ FC FB. ButFD FE∗ FCfor soft semi-open set FE∗, since FC is soft swg-open set in FA. Thus FD FE∗⋂ FE FC⋂ FE FB. It follows that FD 𝑖𝑛𝑡(𝑐𝑙(FB)).Because soft set FBis soft swg-open set. It implies FD int(cl(FB)) wheneverFDis a soft semi-closed set and FD FB. Therefore, FBis soft swg-openset in FA.

IV. SOFT CLOSURE And SOFT SWG-INTERIOR& RESULTS

Definition 4.1.Let (FA, τ )be soft topological space,FB FA. The soft swg-closure of FB(brieﬂy, swg-cl(FB)) to be the intersection of all softswg-closed subsets containing FB. In symbols, swg-cl(FB) = {FC:FB FCand FCis a soft swg-closed set in FA}

Theorem 4.2.While any soft subset FBof soft topological space FA, FB swg-cl(FB ) cl(int(FB))

Proof.It is based on Theorem 2.3.

Example 4.3.We have following example 2.2., FB = {(e1, {c}), (e2, {a, b})},then swg-cl(FB) = {(e1, {b, c}), (e2, {a, b})} and int(FB) = F2, cl(int(FB))= FA. Therefore, FB swg-cl(FB)cl(FB) in FA .

Theorem 4.4.Thesoft swg-closure operator is soft Kuratowski closure operator on FA.

Proof.(i) swg-cl(Fϕ) = Fϕ.

(ii) FB swg-cl(FB) by Theorem 4.2.,

(iii) Let FB1∪ FB2 FCand FCbelongs to softswg-closed set in FA,thenFBi FCand by Deﬁnition 3.1. swg-cl(FB_i) FCfor each i=1, 2. Therefore,swg-cl(FB₁)∪ swg-cl(FB2) ⋂{FC: FB1∪ FB2 FC and FCissoft swg-closed set in FA}=swg-cl(FB₁∪ FB₂). For the reverse insertion,let fA∈swg-cl(FB1∪ FB2) and suppose thatfA ∉swg-cl(FB1)∪ swg-cl(FB2). Then there exist soft swg-closed sets FC₁and FC₂with FB₁ FC₁,fA∉ FC₁ and FB₂ FC₂, fA∉ FC₂. We have FB1∪ FB2 FC1∪ FC2 andFC1∪ FC2is a soft swg-closed set such that fA∉ FC1∪ FC2. Thus fA ∉swg-cl(FB₁∪ FB₂). Which is a contradiction to fA∈ swg-cl(FB1∪ FB2).Henceswg-cl(FB1)∪ swg-cl(FB2

)=swg-cl(FB1∪ FB2 ).

(iv) Let FB FCand FCis soft swg-closed set withinFA. Then byDeﬁnition 3.1.swg-cl(FB) FCand swg-cl(swg-cl(FB)) FC, we haveswg-cl(swg-cl(FB)) ⋂{FC: FB FCand FCis a soft swg-closed set inFA}= swg-cl(FB). [By Theorem 3.2.], swg-cl(FB) swg-cl(swg-cl(FB))and therefore, swg-cl(FB) = swg-cl(swg-cl(FB)). Hence softswg-closure operator is a soft Kuratowski closure operator on FA

Theorem 4.5.While any soft subset FBof soft topological space in FA.

(i) swg-cl(FB) is smallest soft swg-closed set containing FB.

(ii) FBissoft swg-closed set⟺swg-cl(FB) = FB .

Proposition4.6.While two soft subsets FB1and FB2of soft topological space in FA.

(i) FB1 FB2,then swg-cl(FB1)swg-cl(FB2). (ii) swg-cl(FB1⋂ FB2)swg-cl(FB1)⋂swg-cl(FB2).

Theorem 4.7.For any fA∈ FA, fA ∈swg-cl(FB),⟺FC⋂ FB ≠ Fϕfor every soft swg-open set

Proof.Let fA∈swg-cl(FB) for any fA∈ FA. Suppose there exists a softswg-open set FC containing fA such that FC⋂ FB= Fϕ. ThenFB FCc.Since FCcis a soft swg-closed set containing FB, we obtaincl(int(FB)) FCcwhich implies that fA ∉swg-cl(FB), which is a contradiction. On the contrary,assume thatfA∉swg-cl(FB). From Deﬁnition 4.1.there exists a softswg-closed set FDcontaining FBsuch that fA∉FD. Thereafter fA∈ FDcand FDc is a softswg-open set in FA. Also, FDc⋂ FB= Fϕwhich is acontradiction to the hypothesis. Therefore,fA∈swg-cl(FB).

Theorem 4.8.For anyFB FA, soft swg-interior of FB (brieﬂy, swg-int(FB)) is illustrated as the union of the entire soft swg-open sets containing in FB.In symbols, swg-int(FB)= { FC : FC FB and FC is a soft swg-openset in FA}

Remark 4.9.For any soft subset FBof soft topological space FA,

(i)swg-int(FB) is soft swg-open set in FA, Since arbitrary union of every soft swg-open sets in FAis in factsoft swg-open sets inFA.

(ii) swg-int(FB) is the largest soft swg-open set in FA contained in FB.

Theorem 4.10.Let FBrepresentsoft swg-open set in FA, ⟺FB=swg-int(FB).

Proof.LetFBis soft swg-open set. Now, FBbeing soft swg-open set in FA, FBis the largest soft swg-open set in FA. Therefore, FB=swg-int(FB). Conversely, let swg-int(FB) = FB and by deﬁnition, swg-int(FB) is a soft swg-open set. Then it follows that, FBis also a soft swg-open set.

Proposition 4.11.For the two soft subset FB and FCof soft topological space FA, then the following holds.

(i) int(cl(FB))swg-int(FB) FB.

(ii) If FB FC, then swg-int(FB) swg-int(FC). (iii) swg-int(FB⋂ FC) swg-int(FB)⋂swg-int(FC). (iv) swg-int(FB∪ FC)swg-int(FB)∪ swg-int(FC). (v) swg-int(FA) = FA.

(iv) swg-int(Fϕ)= Fϕ .

Remark 4.12.In any soft topological space FA, if swg-int(FB) = swg-int(FC) for subsets FBand FC of FA, then it does not imply that FB = FC.

Example 4.13.We have following example 2.2., Let FB = {(e1, {c})}, FC={(e2, {b})},then swg-cl(FB) = Fϕ and swg-cl(FC) = Fϕ. Therefore,swg-int(FB) = swg-int(FC) but FB≠ FC.

Theorem 4.14.While soft subset FBof soft topological spaceFA, then the following holds.

(i) (swg-int(FB))c = swg-cl(FBc). (ii) swg-int(FB) = (swg-cl(FBc)) c. (iii) swg-cl(FB) = (swg-cl(FBc)) c.

Proof.Let fA∈(swg-int(FB))c. ThenfA∉swg-int(FB) and soevery soft swg-open set FC containing fAis such that FC⊈ FB. That isevery soft swg-open set FCcontaining fAis such that FC⋂ FBc= Fϕ [ByTheorem 4.7.], fA∈swg-cl(FBc) and therefore, (swg-int(FB))c swg-cl(FBc). Conversely, letfA∈swg-cl(FBc). Then [by Theorem 3.4]., everysoft swg-open set FD containing fA is

(5)

everysoft swg-open set FDcontaining fA is such that FD ⊈ FB. This implies by deﬁnition 4.8., fA∉swg-int(FB) implies fA∈(swg-int(FB))candswg-cl(FBc) (swg-int(FB))c. Thus (swg-int(FB))c = swg-cl(FBc). (ii)and (iii) follows from (i).

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