On Separation Axioms in Soft Minimal Spaces

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1 On Separation Axioms in Soft Minimal Spaces R. Gowri 1, S. Vembu 2 Assistant Professor, Department of Mathematics, Government College for Women (Autonomous), Kumbakonam, India 1 Research Scholar, Department of Mathematics, Government College for Women (Autonomous), Kumbakonam, India 2 ABSTRACT: The aim of this paper is to introduce the concept of some separation axioms in soft minimal spaces such as Soft T 0 -space, Soft T 1 -space, Soft T 2 -space, soft regular, and soft normal space which involve only pair of distinct soft points and we study some of their properties. KEYWORDS: Soft T 0 - space, Soft T 1 - space, Soft T 2 -space, Soft semi Hausdroff space, Soft Pseudo Hausdroff space, Soft Uryshon space, soft regular, soft normal, Completely soft normal space. I. INTRODUCTION V. Popa and T. Noiri [10] introduced the concept of minimal structure. Also they introduced the notion of m X - open set and m X -closed set and characterize those sets using m X -closure and m X -interior operators respectively. Further they Introduced M-continuous functions and studied some of its basic properties. Recently, many researchers improved minimal spaces see [1] [2] [3] and. M. Tunapan [13] introduced some separation axioms in biminimal structure spaces such as CT 0 -space, CT 1 -space, C-Hausdroff space, C-regular space and C-normal space. D. Molodtsov [9] introduced the concept of soft set theory as a general mathematical tool to deal with uncertainties while modeling the problems with incomplete information. Many researches improved the concept of soft sets. Shabir and Naz [12] initiated the study of soft topological spaces. They defined soft topology as a collection of soft sets over X. Consequently, they defined basic notions of soft topological spaces such as soft open and soft closed sets, soft subspace, soft closure, soft neighborhood of a point and soft separation axioms are also introduced and their basic properties are investigated by them. In this paper, we introduce and study some separation axioms in soft minimal spaces. II. PRELIMINARIES 2.1. Definition. [9]. Let U be an initial universe and E is a set of parameters. Let P (U) denotes the power set of U and A is a nonempty subset of E. A pair (F, A) is called a soft set over U, where F is a mapping given by F: A P (U) In other words, a soft set over U is a parameterized family of subsets of the universe U. For A. F ( ) may be considered as the set of - approximate elements of the soft set (F, A) Definition. [7]. Let X be an initial universe set, E be the set of parameters and A E. Let F A be a nonempty soft set over X and p (F A ) is the soft power set of F A. A subfamily m of p (F A ) is called a soft minimal set over X if F m and F A m. (F A,m ) or (X,m, E) is called a soft minimal space over X. Each member of m is said to be m -soft open (or soft m -open) set and the complement of an m -soft open (or soft m -open) set is said to be m -soft closed (or soft m -closed) set over X Definition. [7]. Let (F A,m ) be a soft minimal space over X. For a soft subset F B of F A, the m -soft closure (or soft m -closure) of F B and m -soft interior (or soft m -interior) of F B are defined as follows: (1) m Cl (F B ) = {F : F B F, F A -F m }, (2) m Int (F B ) = {F : F F B, F m }. Copyright to IJIRSET DOI: /IJIRSET

2 2.4. Definition: [7]. Let (F A,m ) be a soft minimal space and F Y be a soft subset of F A. Define soft minimal set m FY on F Y as follows: m FY = {F B F Y F B m }. Then (F Y, m FY ) it is called a soft minimal subspace of (F A,m ). III. SOFT T 0 - SPACE, SOFT T 1 - SPACE and SOFT T 2 - SPACE in SOFT MINIMAL SPACES 3.1. Definition: Let (F A,m ) be a soft minimal space over X and (x, u), (y, u) F A such that (x, u) (y, u). If there exist at least one soft m -open set U B or V B such that (y, u) U B or (x, u) V B, then (F A,m ) is called a soft T 0 - space Definition: Let (F A,m ) be a soft minimal space over X and (x, u), (y, u) F A such that (x, u) (y, u). If there exist soft m -open sets U B and V B such that (x, u) U B, (y, u) U B and (y, u) V B, (x, u) V B then (F A,m ) is called a soft T 1 -space Definition: Let (F A,m ) be a soft minimal space over X and (x, u), (y, u) F A such that (x, u) (y, u). If there exist soft m -open sets U B and V B such that (x, u) U B, (y, u) V B, and U B V B = F, then (F A,m ) is called a soft T 2 - space Example: Let U = {u 1, u 2 }, E = {x 1, x 2, x 3 }, A = {x 1, x 2 } E and F A = {(x 1 {u 1, u 2 }), (x 2 {u 1, u 2 })}. Then F A1 = {(x 1 {u 1 })}, F A2 = {(x 1 {u 2 })}, F A3 = {(x 1 {u 1, u 2 })}, F A4 = {(x 2 {u 1 })}, F A5 = {(x 2 {u 2 })}, F A6 = {(x 2 {u 1, u 2 })}, F A7 = {(x 1 {u 1 }), (x 2 {u 1 })}, F A8 = {(x 1 {u 1 }), (x 2 {u 2 })} F A9 = {(x 1 {u 1 }), (x 2 {u 1, u 2 })}, F A10 = {(x 1 {u 2 }), (x 2 {u 1 })} F A11 = {(x 1 {u 2 }), (x 2 {u 2 })}, F A12 = {(x 1 {u 2 }), (x 2 {u 1, u 2 } F A13 = {(x 1 {u 1, u 2 }), (x 2 {u 1 })}, F A14 = {(x 1 {u 1, u 2 }), (x 2 {u 2 })} F A15= F A,, F A16= F are all soft subset of F A, P(F A ) = 2 4 = 16. Take m = {F, FA 2, FA 5, FA 8, FA 10, F A }, then (F A,m ) is soft T 2 -space Proposition: (1) Every soft T 1 -space is soft T 0 -space. (2) Every soft T 2 -space is soft T 1 - space. Proof: Let (F A,m ) be a soft minimal space over X and (x, u), (y, u) F A such that (x, u) (y, u). (1) Obvious (2) Let (F A,m ) is a soft T 2 -space, then by definition of soft T 2 -space (x, u), (y, u) F A such that (x, u) (y, u), if there exist soft m -open sets U B and V B such that (x, u) U B,(y, u) V B and U B V B = F, Since U B V B = F, So (x, u) V B and (y, u) U B. Thus it follows (F A,m ) is a soft T 1 -space. Note that every soft T 1 -space is soft T 0 -space. Every soft T 2 -space is soft T 1 -space. But the converses do not hold in general Example: Let (F A,m ) be a soft minimal space where Let U = {u 1, u 2 }, E = {x 1, x 2, x 3 }, A = {x 1, x 2 } E and F A = {(x 1 {u 1, u 2 }), (x 2 {u 1, u 2 })}. m = {F, FA 7, F A }. Thus (F A,m ) is soft T 0 -space but not soft T 1 -space Example: Let (F A,m ) be a soft minimal space where Let U = {u 1, u 2 }, E = {x 1, x 2, x 3 }, A = {x 1, x 2 } E and F A = {(x 1 {u 1, u 2 }), (x 2 {u 1, u 2 })}. m = {F, FA 2, FA 9, FA 11, FA 13, F A }, Thus (F A,m ) is soft T 1 -space but not soft T 2 -space Lemma: Let (F A,m ) be a soft minimal space and let (F Y, m FY ) be a soft closed subspace of (F A,m ).If U B is soft m open subset of (F A,m ), then U B F Y is also soft m FY -open subset of (F Y, m FY ). Proof: Let U B be a soft m -open subset of (F A,m ). Since (F Y, m FY ) be a soft closed subset of (F A,m ). Then F A -U B is a soft m -closed subset of (F A,m ). Since (F Y, m FY ) be a soft closed subset (F A,m ). Copyright to IJIRSET DOI: /IJIRSET

3 But (F A -U B ) F Y = F Y (U B F Y ). Therefore, F Y - (U B F Y ) is a soft m FY -closed subset of (F Y, m FY ). Hence, (U B F Y ) is a soft m FY -open subset of (F Y, m FY ) Proposition: Let (F Y, m FY ) be a soft minimal subspace of (F A,m ). If (F A,m ) is soft T 0 -space then (F Y, m FY ) is also a soft T 0 -space. Proof: Let (F A,m ) is a soft minimal T 0 -space and (F Y, m FY ) be the soft subspace of (F A,m ). If (x, u),(y, u) F Y such that (x, u) (y, u). Since (F Y, m FY ) (F A,m ), then there exist soft m -open sets U B and V B such that (x, u) U B, (y, u) U B or (y, u) V B, (x, u) V B. Then, (x, u) U B F Y and (y, u) U B F Y.. Similarly, it can be proved that (y, u) V B F Y and (x, u) V B F Y. Thus (F Y, m FY ) is soft T 0 -space Proposition: Let (F Y, m FY ) be a soft minimal subspace of (F A,m ). If (F A,m ) is soft T 1 space, then (F Y, m FY ) is also a soft T 1 -space. Proof: The proof is similar to the proof of proposition Proposition: Let (F Y, m FY ) be a soft minimal subspace of (F A,m ). If (F A,m ) is soft T 2 space, then (F Y, m FY ) is also a soft T 2 -space. Proof: Let (F A,m ) is a soft minimal T 2 -space and (F Y, m FY ) be the soft subspace of (F A,m ). If (x, u),(y, u) F Y such that (x, u) (y, u). Since (F Y, m FY ) (F A,m ), then there exist soft m -open sets U B and V B such that (x, u) U B, (y, u) V B and U B V B = F. Therefore, (x, u) U B F Y, (y, u) V B F Y and (U B F Y ) (V B F Y ) = F. Thus (F Y, m FY ) is soft T 2 -space Proposition: Let (F A,m ) be a soft minimal space over X and (x, u), (y, u) F A such that (x, u) (y, u). If there exist soft m -open sets U B and V B such that (x, u) U B and (y, u) (U B ) C or (y, u) V B and (x, u) (V B ) C, then (F A,m ) is a soft T 0 -space. Proof: Let (y, u) (U B ) C such that (x, u) (y, u) and U B and V B are soft m -open sets such that (x, u) U B and (y, u) (U B ) C or (y, u) V B and (x, u) (V B ) C. If (y, u) (U B ) C. This implies that (y, u) U B. Similarly, we can show that if (x, u) (V B ) C implies (x, u) V B. Hence (F A,m ) is soft T 0 -space Theorem: Let (F A,m ) be a soft minimal space over X and (x, u), (y, u) F A such that (x, u) (y, u). If there exist soft m -open sets U B and V B such that (x, u) U B, (y, u) (U B ) C and (y, u) V B, (x, u) (V B ) C, then (F A,m ) is a soft T 1 -space. Proof: The proof is similar to the proof of proposition Theorem: A soft minimal space (F A,m ) is a soft T 1 -space if and only if for each singleton set {(x, u)} is soft m closed set in F A. Proof: Suppose F A is soft T 1 -space.let (x, u) F A. Let {(y, u)} {(x, u)} C, then (x, u) (y, u) and so there exist soft m -open set U B containing (y, u) but not (x, u) and soft m -open set V B containing (x, u) but not (y, u), (y, u) U B {(x, u)} C. Therefore {(x, u)} C is soft m -open set then {(x, u)} is soft m -closed set. Conversely, suppose (x, u), (y, u) F A, such that (x, u) (y, u). Since {(x, u)} is soft m closed set, then {(x, u)} C is soft m -open set containing (y, u) but not (x, u). Similarly, {(y, u)} C is soft m -open set containing (x, u) but not (y, u). Therefore (F A,m ) is soft T 1 -space Theorem: A soft minimal space (F A,m ) is a soft T 2 -space and for any (x, u), (y, u) F A such that (x, u) (y, u), then there exist soft m -closed sets U B and V B such that (x, u) U B, (y, u) U B and (y, u) V B, (x, u) V B and U B V B = F A. Proof: Since (F A,m ) is a soft T 2 -space and (x, u), (y, u) F A such that (x, u) (y, u), there exist soft m -open sets K B and L B such that (x, u) K B, (y, u) L B and K B L B = F. Clearly K B (L B ) C and L B (K B ) C. Hence, (x, u) (L B ) C. Put (L B ) C = U B. This gives (x, u) U B and (y, u) U B. Also (y, u) (K B ) C. Put (K B ) C = V B. This gives (x, u) V B and (y, u) V B. Therefore, (x, u) U B and (y, u) V B. Hence, U B V B = (L B ) C (K B ) C = F A. Copyright to IJIRSET DOI: /IJIRSET

4 3.16. Definition: A soft minimal space (F A,m ) is said to be soft semi Hausdroff if for every (x, u) (y, u) F A, there exist soft m -open set U B or V B such that (x, u) U B and (y, u) m Cl (U B ) or (y, u) V B and (x, u) m Cl (V B ) Definition: A soft minimal space (F A,m ) is said to be soft Pseudo Hausdroff space if for every (x, u) (y, u) in F A, there exist soft m -open sets U B and V B such that (x, u) U B, (y, u) m Cl (U B ) and (y, u) V B, (x, u) m Cl (V B ) Theorem: A soft minimal space (F A,m ) is soft Pseudo Hausdroff, and then every soft subspace (F Y, m FY ) of (F A,m ) is also soft pseudo Hausdroff. Proof: Let soft minimal space (F A,m ) is a soft pseudo Hausdroff. Let (F Y, m FY ) be the soft subspace of (F A,m ). Since (F A,m ) is a soft pseudo Hausdroff, then for every (x, u) (y, u), there exist soft m -open sets U B and V B such that (x, u) U B, (y, u) m Cl (U B ) and (y, u) V B, (x, u) m Cl (V B ). Then U B F Y and V B F Y are soft m FY - open sets in (F Y, m FY ) such that (x, u) U B F Y and (y, u) m FY Cl (U B F Y ) also (y, u) V B F Y and (x, u) m FY Cl (V B F Y ). Therefore (F Y, m FY ) is soft Pseudo Hausdroff Definition: A soft minimal space (F A,m ) is said to be soft Uryshon space if given (x, u) (y, u) in F A, there exist soft m -open sets U B and V B such that (x, u) U B, (y, u) V B and m Cl (U B ) m Cl (V B ) =F Theorem: In a soft minimal space (F A,m ) is soft Uryshon, then every soft subspace of a soft Uryshon space is soft Uryshon. Proof: Let (F A,m ) is soft Uryshon space and let (F Y, m FY ) be a soft subspace of (F A,m ). Since (F A,m ) is soft Uryshon, then for every (x, u) (y, u), there exist soft m -open sets U B and V B such that (x, u) U B, (y, u) V B and m Cl (U B ) m Cl (V B ) =F. Now U B F Y and V B F Y are soft m FY -open sets such that (x, u) U B F Y and (y, u) V B F Y in (F Y, m FY ). Consider, m FY Cl (U B F Y ) m FY Cl (V B F Y ) = [ m FY Cl (U B F Y )] [m FY Cl (V B F Y )] = [m FY Cl (U B ) m FY Cl (V B )] F Y = [m Cl (U B ) m Cl (V B )] F Y Hence (F Y, m FY ) is soft Uryshon space. = F F Y = F. IV. SOFT REGULAR T 3 -SPACE, SOFT NORMAL T 4 -SPACE IN SOFT MINIMAL SPACES In this section, we introduce soft regular T 3 -space, soft normal T 4 -space and completely soft normal space Definition: Let (F A,m ) be soft minimal space. A soft minimal space (F A,m ) is said to be soft regular if for each soft point (x, u) F A and each soft m -closed set G B not containing (x, u), there exist soft m -open sets U B and V B such that (x, u) U B, G B V B and U B V B = F Example: Let (F A,m ) be a soft minimal space where Let U = {u 1, u 2 }, E = {x 1, x 2, x 3 }, A = {x 1, x 2 } E and F A = {(x 1 {u 1, u 2 }), (x 2 {u 1, u 2 })}. m = {F, F A3, F A6, F A7, F A11, F A }. Take G B = F A6, F A1, G B, then there exists F A3 and F A6 are soft m -open sets such that F A1 F A3, F A6 F A6, F A6 F A6 and F A3 F A6 = F and similarly the other cases. Hence (F A,m ) is soft regular space. Copyright to IJIRSET DOI: /IJIRSET

5 4.3. Proposition: Let (F A,m ) be a soft minimal space, G B be soft m -closed set and (x, u) F A such that (x, u) G B. If (F A,m ) is soft regular space, then there exist soft m -open set U B such that (x, u) U B and U B G B = F. Proof: It is obvious from Definition Proposition: Let (F A,m ) be a soft minimal space and (x, u) F A. If (F A,m ) is soft regular space, then: (1) (x, u) G B if and only if {(x, u)} G B. (2) If {(x, u)} G B = F, then (x, u) G B. Proof: It is obvious Theorem: Let (F A,m ) be a soft minimal space and (x, u) F A. If (F A,m ) is soft regular space, then: (1) For a soft m -closed set G B, (x, u) G B if and only if {(x, u)} G B = F. (2) For a soft m -open set V B, (x, u) V B if and only if {(x, u)} V B = F. Proof: (1) Suppose (F A,m ) be a soft regular space and (x, u) F A. Let G B be a soft m -closed set such that (x, u) G B. Then by proposition 4.3, there exist soft m -open set U B such that (x, u) U B and U B G B = F. It follows that {(x, u)} V B from proposition 4.4 (1). Hence {(x, u)} G B = F. Conversely, if {(x, u)} G B = F, then (x, u) G B from Proposition 4.4 (2). (2) Let V B be soft m -open set such that (x, u) V B. Assume (x, u) V B. This means that, {(x, u)} V B = F. Hence (V B ) C is soft m -closed set such that {(x, u)} (V B ) C. It follows by (1) {(x, u)} (V B ) C = F. This implies that {(x, u)} V B and so (x, u) V B, which is contradiction with (x, u) V B. Therefore, {(x, u)} V B = F. Conversely, if {(x, u)} V B = F, then it is obvious that (x, u) V B. This completes the proof Corollary: Let (F A,m ) be a soft minimal space and (x, u) F A. If (F A,m ) is soft regular space then the following are equivalent: (1) (F A,m ) is soft T 1 -space. (2) (x, u) (y, u) F A such that (x, u) (y, u), there exist soft m -open sets U B and V B such that {(x, u)} U B and {(y, u)} U B = F and (y, u) V B and {(x, u)} V B = F. Proof: It is obvious from Theorem Theorem: Let (F A,m ) be a soft minimal space and (x, u) F A. Then the following are equivalent: (1) (F A,m ) is soft regular space. (2) For every soft m -closed set G B such that {(x, u)} G B = F, there exist soft m -open sets U B and V B such that {(x, u)}, U B, G B V B and U B V B =F. Proof: (1) (2) Let G B be a soft m -closed set such that {(x, u)} G B =F.Then (x, u) G B from Theorem 4.5 (1). It follows by (1), there exist soft m -open sets U B and V B such that (x, u) U B, G B V B and U B V B = F. This means that, {(x, u)} U B, G B V B and U B V B = F. (2) (1) Let G B be a soft m -closed set such that (x, u) G B. Then {(x, u)} G B = F from Theorem 4.5(1). It follows by (2), there exist soft m -open sets U B and V B such that {(x, u)} U B, G B V B and U B V B = F. Hence {(x, u)} U B, G B V B and U B V B = F. Thus, (F A,m ) is soft regular space Theorem: Let (F A,m ) be a soft minimal space, then (F A,m ) is soft regular if and only if for each soft m -open set U B and (x, u) U B, there exist soft m -open set V B such that (x, u) V B, m Cl(V B ) U B. Proof: Let (F A,m ) be soft regular space. Let (x, u) U B, where U B is soft m -open. Let H B = (U B ) C, then H B is soft m -closed, (x, u) H B. By the soft regularity of (F A,m ), there exist soft m -open sets N B and M B such that (x, u) M B, H B N B and M B N B = F. Then M B (N B ) C, m Cl (M B ) m Cl (N B ) C = (N B ) C. (1) H B (N B ) C, then (N B ) C (H B ) C = U B, then (N B ) C U B (2) From (1) and (2), we have (x, u) M B, m Cl (M B ) U B. Copyright to IJIRSET DOI: /IJIRSET

6 Conversely, let H B be soft m -closed set and (x, u) H B. Let U B = (H B ) C, then U B is soft m -open and (x, u) U B. By hypothesis, there exist soft m -open set M B such that (x, u) M B, m Cl (M B ) U B, H B [m Cl (M B )] C. Since (x, u) M B, M B [m Cl (M B )] C = F. Hence (F A,m ) is soft regular Theorem: A soft minimal space (F A,m ) is soft regular space if and only if for each (x, u) F A and soft m -closed set G B such that (x, u) G B, there exist soft m -open set U B such that (x, u) U B and m Cl (U B ) G B = F. Proof: Suppose that (F A,m ) is a soft regular space. Let (x, u) F A and G B be a soft m -closed such that (x, u) G B. Then there exist soft m -open sets U B and V B such that (x, u) U B, G B V B and U B V B = F. Now U B V B = F. U B (V B ) C and G B V B, (V B ) C (G B ) C. That implies U B (V B ) C (G B ) C. This implies that m Cl (U B ) m Cl (V B ) C = (V B ) C (G B ) C. Therefore m Cl (U B ) G B = F. Conversely, suppose (x, u) F A and G B be soft m -closed set such that (x, u) G B. Then by hypothesis there exist a soft m -open set U B such that (x, u) U B and m Cl (U B ) G B = F. Now m Cl (U B ) G B = F implies that G B [m Cl (U B )] C. Also U B m Cl (U B ) implies [m Cl (U B )] C (U B ) C. U B [m Cl (U B ) C ] C. Therefore U B [m Cl (U B )] C = F. Then there exist soft m -open sets U B and m Cl (U B ) C such that (x,u) U B, G B [m Cl(U B )] C and U B [m Cl(U B )] C = F. Hence (F A,m ) is soft regular space Definition: Let (F A,m ) be a soft minimal space. Then (F A,m ) is said to be a soft T 3 -space, if it is a soft regular and a soft T 1 -space Theorem: Every soft T 3 -space is a soft T 2 -space. Proof: Let (F A,m ) be a soft T 3 -space and (x, u), (y, u) F A such that (x, u) (y, u). Since (F A,m ) is a soft T 1 - space, and {(y, u)} is soft m -closed and (x, u) {(y, u)}. It follows from soft regularity, there exist soft m open sets U B and V B such that (x, u) U B, {(y, u)} V B and U B V B = F. Thus (x, u) U B, (y, u) {(y, u)} V B and U B V B = F. Therefore, (F A,m ) is soft T 2 -space Theorem: A soft subspace (F Y, m FY ) of a soft T 3 -space (F A,m ) is soft T 3 -space. Proof: First we want to prove that (F Y, m FY ) is soft T 1 -space.if (x, u), (y, u) F Y such that (x, u) (y, u). Since (F Y, m FY ) (F A,m ), then there exist soft m -open sets U B and V B such that (x, u) U B, (y, u) U B and (y, u) V B, (x, u) V B. Therefore, (x, u)=u B F Y and (y, u) U B F Y and (y, u) V B F Y and (x, u) V B F Y. Thus (F Y, m FY ) is soft T 1 -space. Next we want to prove (F Y, m FY ) is soft regular space. Let (y, u) F Y and H B be soft m FY -closed set in (F Y, m FY ) such that (y, u) H B. Then H B = G B F Y, where G B is soft m -closed set in (F A,m ). Hence, (y, u) G B F Y. But (y, u) F Y, so (y, u) G B. Since (F A,m ) is soft T 3 -space, so there exist soft m -open sets U B and V B in (F A,m ) such that (y, u) U B, G B V B and U B V B = F. Take M B =U B F Y and N B =V B F Y, then U B F Y and V B F Y are soft m FY -open sets in (F Y, m FY ) such that (y, u) M B, H B V B F Y =N B and M B N B U B V B = F. Thus, (F Y, m FY ) is soft T 3 -space Definition: A soft minimal space (F A,m ) is said to be soft normal if for each pair of soft m -closed sets G B and H B in F A and such that G B H B = F, there exists soft m -open sets U B and V B such that G B U B, H B V B and U B V B = F Theorem: A soft minimal space (F A,m ) is soft normal if and only if for every soft m -closed set G B in F A and soft m -open set U B in F A containing G B, there exist soft m -open set V B such that G B V B m Cl(V B ) U B. Proof: Suppose (F A,m ) is soft normal space, let G B be soft m -closed set in F A and U B is soft m -open in F A such that G B U B. Then (U B ) C is soft m -closed in F A and G B (U B ) C = F. So there exists soft m -open sets V B and K B such that (U B ) C K B, G B V B and V B K B = F, (K B ) C U B, V B (K B ) C. This implies that m Cl (V B ) m Cl (K B ) C =(K B ) C.Then G B V B m Cl (V B ) U B. Copyright to IJIRSET DOI: /IJIRSET

7 Conversely, let G B and H B be soft m -closed sets in F A such that G B H B = F, then (H B ) C is soft m -open set in F A, and G B (H B ) C. By hypothesis, there exist soft m -open set V B such that G B V B, m Cl (V B ) (H B ) C, then H B [m Cl (V B )] C. So we have G B V B, H B [m Cl (V B )] C and V B [m Cl (V B )] C =F. Therefore, (F A,m ) is soft normal space Theorem: A soft closed subspace of a soft normal space is soft normal. Proof: Let (F A,m ) be a soft normal space. Let (F Y, m FY ) be soft closed subspace of (F A,m ). Let M B and N B be two soft disjoint soft m FY closed subsets in (F Y, m FY ).Then there exists soft m -closed sets G B and H B in F A such that M B = G B F Y, N B = H B F Y. Hence by soft normality there exist soft m -open sets U B and V B such that M B U B, N B V B and U B V B =F. Since M B, N B F Y, then M B U B F Y, N B V B F Y and ( U B F Y ) (V B F Y )=F, where U B F Y and V B F Y are soft m FY -open sets in F Y. Therefore, (F Y, m FY ) is soft normal space Definition: Let (F A,m ) be a soft minimal space. Then (F A,m ) is said to be soft T 4 -space, if it is a soft normal and soft T 1 - space Proposition: Every soft T 4 -space is also soft T 3 -space Proof: Let (F A,m ) be a soft T 4 -space, then (F A,m ) is soft normal as well as soft T 1 -space. To prove that the space is soft T 3 -space, it suffices to show that the space is soft regular. Let K B be a soft m -closed soft subset of F A and let (x, u) be a soft point of F A, such that (x, u) K B. Since (F A,m ) is soft T 1 -space.thus {(x, u)} is a soft m -closed soft subset of F A, such that {(x, u)} K B =F, then by soft normality, there exist soft m -open sets U B and V B in F A such that {(x, u)} U B and K B V B and U B V B =F. Also {(x, u)} U B, then (x, u) U B, then there exist soft m -open sets U B and V B such that {(x, u)} U B, K B V B and U B V B =F. It follows that the space (F A,m ) is soft regular and so soft T 3 -space Remark: Every soft T 4 -space is also soft T 3 -space; every soft T 3 -space is also soft T 2 -space Definition: A soft minimal space (F A,m ) is said to be completely soft normal if for each pair of soft disjoint soft m -closed sets U B and V B F A, there exist soft m -open sets K B and N B such that U B K B, V B L B and m Cl(K B ) m Cl(L B )=F Theorem: If (F A,m ) is completely soft normal then it is soft normal. Proof: Let (F A,m ) is completely soft normal minimal space. Let U B and V B are any two soft disjoint soft m closed sets. Then by completely soft normality, we have there exist soft m -open sets K B and L B such that U B K B, V B L B and K B L B =F Theorem: If (F A,m ) is completely soft normal then every soft subspace is soft normal. Proof: Let (F A,m ) is completely soft normal and (F Y, m FY ) be soft subspace. Let M B and N B are two soft disjoint soft m FY -closed sets in (F Y, m FY ) Then M B = U B F Y and N B = V B F Y, where U B and V B are soft m -closed sets in (F A,m ). Then there exist soft m open sets K B and L B in F A such that U B K B, V B L B and K B L B =F. This implies U B F Y K B F Y, V B F Y L B F Y and (K B F Y ) (L B F Y ) =F. Thus, (F Y, m FY ) is soft normal. V. CONCLUSION In this paper, we introduce the notion of separation axioms in soft minimal spaces. In particular, we study the properties of soft T i, i = 0, 1, 2, soft regular and soft normal spaces. We hope that, the results in this paper will help researcher enhance and promote the further study on soft minimal to carry out general framework for their application in practical life. Copyright to IJIRSET DOI: /IJIRSET

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