Nano Generalized Pre Homeomorphisms in Nano Topological Space

Transcription

1 International Journal o Scientiic esearch Publications Volume 6 Issue 7 July Nano Generalized Pre Homeomorphisms in Nano Topological Space K.Bhuvaneswari 1 K. Mythili Gnanapriya 2 1 Associate Proessor Department o Mathematics Mother Teresa Women s niversity Kodaikanal Tamilnadu India. 2 esearch Scholar Department o Mathematics Karpagam Academy o Higher Education oimbatore Tamilnadu India. Abstract- The purpose o this paper is to deine study some properties o Nano generalized pre-homeomorphisms in Nano topological spaces. Index Terms- Nano Topology Ngp-closed sets Ngp-closed map Ngp-open map Ngp- homeomorphisms T I. INTODTION he notion o homeomorphism plays a very important role in topology. Many researchers have generalized the notions o homeomorphism in topological spaces. Maki et al.[5] have introduced investigated g-homeomorphism gc - homeomorphism in topological spaces. ellis Thivagar [4] introduced Nano homeomorphisms in Nano Topological Spaces. Bhuvaneswari et al. [2] introduced studied some properties o Nano generalized homeomorphisms in Nano topological spaces. In this paper a new cls o homeomorphism called Nano generalized pre homeomorphism is introduced some o its properties are discussed. Deinition 2.1. A bijective unction is both g - continuous g - open. Deinition 2.2. A bijective unction irresolute unctions. II. PEIMINAIES : ( X ) ( Y ) is called a generalized homeomorphism (g-homeomorphism) [5] i : ( X ) ( Y ) is called a gc - homeomorphism [5] i both -1 are gc - Deinition 2.3. [6] et be a non-empty inite set o objects called the universe be an equivalence relation on named the indiscernibility ( ) relation. Elements belonging to the same equivalence cls are said to be indiscernible with one another. The pair is said to be the approximation space. et X. (i) The lower approximation o X with respect to is the set o all objects which can be or certain clsiied X with respect to ( X ) its is denoted by. ( X ) ( : ( X That is xu where ( denotes equivalence cls determined by x. (ii) The upper approximation o X with respect to is the set o all objects which can be possibly clsiied X with respect to ( X ) ( : ( X (X ) it is denoted by.that is xu (iii) The boundary region o X with respect to is the set o all objects which can be clsiied neither X nor not-x with respect B ( X ) to it is denoted by B ( X ) ( X ) ( X ). That is Property 2.4. [6] ( ) I is an approximation space X Y then ( X ) X ( X ) (i) ( ) ( ) (ii) ( ) ( )

2 International Journal o Scientiic esearch Publications Volume 6 Issue 7 July (iii) (iv) (v) (vi) (vii) (viii) (i ( ( X Y ) ( X ) ( Y ) ( X Y ) ( X ) ( Y ) ( X Y ) ( X ) ( Y ) ( X Y ) ( X ) ( Y ) ( X ) ( Y ) ( X ) ( Y ) whenever ( X ) [ ( X )] ( X ) [ ( X )] ( X ) ( X ) ( X ) ( X ) ( X ) ( X ) X Y Deinition 2.5. [3] et be the niverse be an equivalence relation on ( X ) { ( X ) ( X ) B ( X )} where X by property 2.4. (X ) satisies the ollowing axioms: (i) ( X ). (X ) (ii) The union o the elements o any sub collection o (X ) is in (X ) (iii) The intersection o the elements o any inite sub collection o is in ( X ). (X ) That is is a topology on called the Nano topology on with respect to X. ( We call (X ) the Nano topological space. The elements o are called Nano-open sets. The elements o the (X ) complement o are called Nano-closed sets. Deinition 2.6. [4] A map is said to be Nano closed map (resp. Nano open map ) i the image o every Nano closed set (resp. Nano open set) in is Nano closed (resp. Nano open) in Deinition 2.7. [4] A unction is Nano continuous is Nano open. Deinition 2.8. [2] A bijection ( V ( is called Nano generalized homeomorphism (shortly Nghomeomorphism) i is both Ng-continuous Ng-open. ( V ( is said to be Nano homeomorphism i is 1-1 onto III. POPETIES OF NANO GENEAIZED PE HOMEOMOPHISM IN NANO TOPOOGIA SPAE Deinition 3.1. A map is said to be Nano generalized pre-closed map (shortly- Ngp-closed map) i the image o every Nano closed set in is Ngp-closed in. Deinition 3.2. A map is said to be Ngp*-closed map i the image ( V ( or every Ngp-closed set A in. Deinition 3.3. A bijection : ( V ( is called Nano generalized pre homeomorphism (shortly Ngphomeomorphism) i is both Ngp-continuous Ngp-open. (A) is Ngp-closed in

3 International Journal o Scientiic esearch Publications Volume 6 Issue 7 July Example 3.4. et = {a b c d} with = {{a} {c} {b d}} X = {a b} (X ) = { Ф {a} {a b d} {b d} }.Then = { Ф {a b d} {a b}{d} V } with V = {{d}{c}{a b}} Y = {a d}. Deine ( V ( ( a) b ( a ( c ( d) d is bijective Ngp-continuous Ngp-open. Thereore is an Ngp-homeomorphism. Theorem 3.5. et only i is Ngp-closed Ngp-continuous. ( V ( be a one to one onto mapping is an Ngp-homeomorphism i Proo. et be an Ngp-homeomorphism is Ngp-continuous. et B be an arbitrary Nano closed set in B is Ngp-closed set in B is Ngp-open.Since is Ngp-open ( B) is Ngpopen in. That is V (B) is Ngp-open in. Thereore (B) is Ngp-closed in.thus the image o every Nano closed set in is Ngp-closed in onversely let be Ngp-closed Ngp-continuous. et B be a Nano open set in B is Nano closed in. Since is Ngp-closed ( B) V ( B) is Ngp-closed in V. Thereore (B) is Ngp-open in ( V (. Thus is Ngp-open hence is an Ngp- homeomorphism. Theorem 3.6. et ( V ( be a bijective Ngp-continuous map the ollowing are equivalent. (i) is an Ngp-open map (ii) is an Ngp-homeomorphism (iii) is an Ngp-closed map. Proo. (i) (ii) By the given hypothesis the map is bijective Ngp-continuous Ngp-open. Hence is Ngp-homeomorphism. (ii) (iii) et A be the Nano closed set in A is Nano open in. By ( A ) sumption is Ngp-open in ( A ) ( ( A)). i.e. is Ngp-open in hence (A) is Ngp-closed in or every Nano closed set A in. Hence the unction is Ngp-closed map. (iii) (i) et F be a Nano open set in F is Nano closed set in. By the ( F ) given hypothesis is Ngp-closed in ( F ) ( ( F)). But is Ngp-closed i.e. (F) is Ngp-open in or every Nano open set F in. Hence is Ngp-open map. Theorem 3.7. Every Nano homeomorphism is Ngp-homeomorphism but not conversely. Proo : I is Nano homeomorphism by deinition is bijective Nano continuous Nano open is Ngp-continuous Ngp-open respectively. Hence the unction ( V ( is an Ngp-homeomorphism. Thereore every Nano homeomorphism is Ngp-homeomorphism.

4 International Journal o Scientiic esearch Publications Volume 6 Issue 7 July The map in Example 3.4 is an Ngp-homeomorphism but not a Nano homeomorphism because it is not Nano continuous Nano open map. Theorem 3.8. Every Ng- homeomorphism is Ngp-homeomorphism but not conversely. Proo : Since every Ng-continuous map is Ngp-continuous every Ng-open map is Ngp-open the theorem ollows. Example 3.9. et (X ) = {Ф {a} {b c} } with = {{a}{b c}} X ={a c } = {Ф {a c} V} with V = {{a c} {b}} Y = {a c}. Deine ( V ( ( a) c ( a ( b which is an Ngp-homeomorphism. But this is not Ng-homeomorphism because or the Nano open set {b c} in ( b { a b} is not Ng-open in. emark The composition o two Ngp-homeomorphisms need not be an Ngp-homeomorphism seen rom the ollowing example. Example et = {a b c} = V = W (X ) = {Ф {b} {a c} } with = {{b} {a c}} X = {b c}. et = {Ф {a} {b c} V} with V (z) = {{a} {b c}} Y = {a c}. et = {Ф {a c} W} with W = {{a} {b c}} Z = {a} Deine ( a) a ( c ( b g : ( V. Deine ( ( W ( Z)) a) c a b g are Ngp-homeomorphisms but their composition ( g ) : ( V ( is not an Ngp- homeomorphisms because or the Nano open set {b} in ( g ) ( W ( Z)) ({b}) = {b} which is not Ngp-open set in.thereore ( g ) is not an Ngp-open map ( g ) is not an Ngp-homeomorphism. We next introduce a new cls o maps called Ngp*-homeomorphisms which orms a subcls o Ngp-homeomorphisms. This cls o maps is closed under composition o maps. 1 Deinition A bijection is said to be Ngp*-homeomorphism i both are Ngp-irresolute. We denote the amily o all Ngp-homeomorphism (resp. Ngp*-homeomorphism) o a Nano topological space onto Ngp h itsel by Ngp * h( (resp. Theorem Every Ngp*-homeomorphism is Ngp-homeomorphism. i.e. For any space Ngp * h( Ngp h(. Proo. Since every Ngp-irresolute unction is Ngp-continuous every Ngp*-open map is Ngp-open the theorem ollows. emark The converse o the above theorem is not true shown rom the ollowing example. Example et = {a b c} = V = W (X ) = {Ф {b} {a c} } with = {{b} {a c}} X = {b c}. et V (z) = {Ф {a} {b c} V} with = {{a} {b c}} Y = {a c}. et W = {Ф {a c} W} with = {{a} {b c}} g : ( V Z = {a} Deine ( ( W ( Z)) a) c a b g is an Ngphomeomorphism but not Ngp*-homeomorphism since or the Ngp-closed set {a b} in 1 1 ( ({ }) g ) a b = a { a c} ( W ( Z)) is not Ngp-closed in.thereore g is not Ngp-irresolute thereore g is not Ngp*- homeomorphism.

5 International Journal o Scientiic esearch Publications Volume 6 Issue 7 July Example Ngp*-homeomorphism Nano homeomorphism et = {a b c} = V = W (X ) = {Ф {b} {a c} } with = {{b} {a c}} X = {b c}. et = {Ф {a} {b c} V} with V (z) = {{a} {b c}} Y = {a c}. et = {Ф {a c} W} with W = {{a} {b c}} Z = {a}. Deine ( a) a ( c ( b is Ngp*-homeomorphism but not Nano homeomorphism because it is not Nano continuous. Example Ng-homeomorphism Ngp*-homeomorphism et (X ) = {Ф {a} {b c} } = {Ф {a} V}. Deine ( V ( ( a) b ( a ( c.then is Ng -homeomorphism or any Ngp-closed set {b} in 1 1 ( ) ({ b}) = ( a is not Ngp-closed in. Thereore 1 is not Ngp-irresolute thereore is not Ngp*- homeomorphism. emark We obtain the ollowing implications rom the above discussions. Figure Ngc-homeomorphisms Nano homeomorphisms Ngp-homeomorphism Ng-homeomorphism Ngp*-homeomorphism Figure 3.19 Implications o the esults EFEENES [1] BhuvaneswariK. K.Mythili Gnanapriya On Nano Generalized Pre closed sets Nano pre Generalized losed sets in Nano Topological spaces. International Journal o Innovative esearch in Science Engineering Technology. 3 (10) : [2] BhuvaneswariK. K.Mythili Gnanapriya On Nano Generalized continuous unction in Nano Topological Spaces. International Journal o Mathematical Archive. 6 (6) : [3] ellis Thivagar M. armel ichard2013. On Nano orms o weakly open sets. International Journal o Mathematics Statistics Invention. 1(1) : [4] ellis Thivagar M. armel ichard On Nano ontinuity. Mathematical Theory Modeling. 3(7): 32 [5] Maki H. P.Sundaram K. Balachran On generalized homeomorphisms in topological spaces. Bull. Fukuoka niv. Ed. Part III. 40: [6] Pawlak Z ough Sets. International Journal o Inormation omputer Science. 11(5): ATHOS First Author Dr. K. Bhuvaneswari Associate Proessor Department o Mathematics Mother Teresa Women s niversity Kodaikanal Tamilnadu India. drkbmaths@gmail.com Second Author K. Mythili Gnanapriya Assistant Proessor Department o Mathematics Nehru Arts Science ollege oimbatore. k.mythilignanapriya@gmail.com