A view on rb-convergence in multiset topological spaces

This paper deals with the concepts of regular b-closed M-sets and regular b-continuous M-set functions are studied with necessary examples. Also the concepts of rb-convergence to a point, rb-accumulates to a point, rb-S-closed M-spaces are studied and some interesting properties are discussed. Finally, sequence of rb-convergence and rb-Hausdorff M-spaces are discussed.

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Introduction

The notion of M-topological space and the concept of open M-sets are introduced by Girish and sunil Jacob John [4]. Moreover, topologies on multisets can be associated to IC-bags or nk- bags introduced by Chakrabarthy [3] (Chakrabarty, 2000; Chakrabarty and Despi, 2007) with the help of rough set theory. The association of rough set theory and topologies on multisets through bags with interval counts (Chakrabarty and Despi, 2007) can be used to develop theoretical study of covering based rough sets with respect to universe as multisets. Nagaveni and Narmadha [9, 10] introduced the notions of regular b-closed (rb-closed) sets in topological space.

Regular open sets and b-open sets have been introduced and investigated by Stone [12] and Andrijevi′c [1]. Generalized α-closed sets have been introduced and investigated Maki, Devi and Balachandran [8]. 2. Preliminaries Definition 2.1 [4]: Let M ∈ [X]w and τ ⊆ P∗ (M). Then τ is called a Multiset topology of M if τ satisfies the following properties. 1.

The M-set M and the empty M-set ϕ are in τ. 2. The M-set union of the elements of any sub collection of τ is in τ. 3. The M-set intersection of the elements of any finite sub collection of τ is in τ. Definition 2.2 [4] Given a subM-set A of an M-topological space M in [X]w, the interior of A is denoted by Int (A) and it is is defined as follows: Int(A) =  {G ⊆ M: G is an open M-set and G ⊆ A} and }.

G , A G : ) x ( C { max ) x ( C G ) A ( Int     Definition 2.3 [4]: Given a subM-set A of an M-topological space M in [X]w, the closure of A is denoted by Cl (A) and is defined asfollows: Cl(A) = {K ⊆ M: K is a closed M-set and A ⊆ K} and }. K , K A : ) x ( C { min ) x ( C c K ) A ( Cl     3. On rb-Closed M-Sets And rb-Continuous M-Set Functions Definition 3.1: Let (M, τ) be an M-topological space in [X] W. Any subM-set A of M is called a regular open M-set if A = int (cl (A)) with CA(x) = Cint (cl (A)) (x), for all x ∈ X.

Maths 2018; 3(3): 226-230 B Amudhambigai Assistant Professsor of Mathematics, Department of Mathematics Sri Sarada College for Women Salem, Tamil Nadu, India G Vasuki Associate Professor of Mathematics, Department of Mathematics Sri Sarada College for Women Salem, Tamil Nadu, India N Santhiya Research Scholar, Department of Mathematics Sri Sarada College for Women Salem, Tamil Nadu, India Correspondence B Amudhambigai Assistant Professsor of Mathematics, Department of Mathematics Sri Sarada College for Women Salem, Tamil Nadu, India Definition 3.2: Let (M, τ) be an M-topological space in [X] W. Any subM-set A of M is called a regular closed M-set if A = cl (int (A)) with CA(x) = Ccl (int (A)) (x), for all x ∈ X. Definition 3.3: Let (M, τ) be an M-topological space in [X] W and A be any subM-set of M. The M-set union of all regular open M-sets of (M, τ) contained in A is called the regular interior of A and is denoted by rint (A) i.e. rint(A) = ⋃ {G ⊆ M: each G is a regular open M-set and G ⊆ A} with Crint (A) (x) = Max {CG(x): G ⊆ A, each G is a regular open M-set}, for all x ∈ X.

Definition 3.4: Let (M, τ) be an M-topological space in [X]W and A be any subM-set of M. The M-set intersection of all regular closed M-sets of (M, τ) containing A is called the regular closure of A and it is denoted by rcl (A). i.e. rcl(A) = ⋂ { K ⊆ M: each K is a regular closed M-set and A ⊆ K } with Crcl(A)(x) = Min {Ck(x): A ⊆ K, each K is a regular closed M-set}, for all x ∈ X. Definition 3.5: Let (M, τ) be an M-topological space in [X] W. Any subM-set A of M is called a b-open M-set if A ⊆ int(cl(A)) ⋃ cl(int(A)) with CA(x) ≤ Max { Cint(cl(A) (x), C Definition 3.11: Let (M, τ) be an M-topological space in [X] W.

Any subM-set A of M is called a generalized α-closed (gα- closed) M-set if αcl(A) ⊆ U with Cαcl(A)(x) ≤ CU(x), whenever A ⊆ U with CA(x) ≤ CU(x), and U is an α-open M-set in (M, τ), for all x ∈ X. Proposition 3.1.1: Every regular closed M-set is a rb-closed M-set. Remark 3.1.2: the converse of the above Proposition 2.1.1 need not be true as shown in the Example 2.1.12. Example 3.1.3 Let X = {a, b, c}, W = 2.

Let M = {1/a, 2/b, 1/c}. Let τ = { M, ϕ, { 1/a }, { 2/b }, { 1/a, 2/b } }. Then τc = {ϕ, M, {2/b, 1/c}, {1/a, 1/c}, {1/c}}. Clearly, τ is an M- topology and the ordered pair (M, τ) is an M-topological space.

Also, the collection of the regular closed M-sets of (M, τ) is { M, ϕ, { 2/b, 1/c }, { 1/a, 1/c } } and the collection of the b-open M-sets of (M, τ) is { ϕ, M, { 1/a }, { 2/b }, { 1/b }, { 1/a, 2/b }, { 2/b, 1/c }, { 1/a, 1/c }, { 1/b, 1/a }, { 1/a, 1/b, 1/c } }. Let A = {1/c} be a sub M-set of M and A ⊂ U = {1/a, 1/c} is the b-open M-set in (M, τ). Then rcl (A) = {1/c} ⊂ U. Therefore, rcl(A) ⊂ U with Crcl(A) (x) < CU(x), for all x ∈ X. whenever A ⊆ U and U is b-open M-set with CA(x) ≤ CU(x), for all x ∈ X.

Hence, A is a rb-closed M-set. But int (A) = ϕ and cl (int (A)) = ϕ. Thus, A ≠ cl (int (A)) with CA(c) = 1 ≠ 0 = Ccl(int(A))(c), CA(a) = Ccl(int(A))(a) = 0 and CA(c) = Ccl(int(A))(c) = 0, for all x ∈ X. Hence, A is not a regular closed M-set.

Proposition 3.1.4: Every regular closed M-set is a gα-closed M-set. Remark 3.1.5: The converse of the Proposition 2.1.2 need not be true as shown in the Example 2.1.13 Example 3.1.6: Let X = {a, b, c}, W = 2. Let M = {1/a, 2/b, 1/c}. Let τ = { ϕ, M, { 1/a }, { 1/a, 2/b }, { 2/b } }.

Then τc = {M, ϕ, {2/b, 1/c}, {1/c}, {1/a, 1/c}}. Clearly, τ is an M- topology and the ordered pair (M, τ) is an M-topological space. Now the collection of the α-open M-sets of (M, τ) is { M, ϕ, { 1/a }, { 1/a, 2/b }, { 2/b } } and α-closed M-sets of (M, τ) is { M, ϕ, { 2/b, 1/c }, { 1/c }, { 1/a, 1/c } }. Let A = {1/b, 1/c} be a sub M-set of M.

Then A ⊂ U = M is the α-open M-set in (M, τ). Then αcl (A) = {2/b, 1/c} ⊂ U. Therefore, αcl (A) ⊂ U with Cαcl (A) (x) < CU(x), for all x ∈ X whenever A ⊂ U and U is an α-open M-set. Hence, A is a gα- closed M-set.

But int (A) = ϕ and cl (int (A)) = ϕ. Thus A ≠ cl(int(A)) with CA(c) = 1 ≠ 0 = Ccl(int(A))(c), CA(a) = Ccl(int(A))(a) = 0 and CA(b) = Ccl(int(A))(b) = 0. Hence, A is not a regular closed M-set. Proposition 3.1.7: Every rb-closed M-set is a gα-closed M- set.

Remark 3.1.8: The converse of the Proposition 2.1.3 need not be true as shown in the Example 2.1.14 Example 3.1.9: Let X = {a, b, c}, W = 2. Let M = {1/a, 2/b, 1/c}. Let τ = { ϕ, M, { 1/a }, { 1/a, 2/b }, { 2/b } }. Then τc = { M, ϕ, { 2/b, 1/c }, { 1/c }, { 1/a, 1/c } }.

Now τ is an M- topology and the ordered pair (M, τ) is an M-topological space. Now the collection of the regular-closed M-sets of (M, τ) is { ϕ, M, { 1/c, 2/b }, { 1/c, 1/a } }, the collection of the α- closed M-sets of (M, τ) is { M, ϕ, { 2/b, 1/c }, { 1/c }, { 1/a, cl(int(A))(x) }, for all x ∈ X. Definition 3.6: Let (M, τ) be an M-topological space in [X]W. Any subM-set A of M is called a regular b-closed (rb-closed) M-set if rcl(A) ⊆ U with Crcl(A)(x) ≤ CU(x), whenever A ⊆ U with CA(x) ≤ CU(x) and U is a b-open M-set in (M, τ), for all x ∈ X.

The complement of a regular b-closed M-set is called a regular b-open (rb-open) M-set. Definition 3.7: Let (M, τ) be an M-topological space in [X] W. For any subM-set A of M, the rb-interior of A is defined as the M-set union of all rb-open M-sets of (M, τ) contained in A and is denoted by rb-int(A) i.e. rb-int(A) = ⋃ { G ⊆ M: each G is a rb-open M-set and G ⊆ A } with Crb-int(A)(x) = Max { CG(x): G ⊆ A, each G is a rb-open M-set }, for all x ∈ X. Definition 2.8: Let (M, τ) be an M-topological space in [X] W.

For any subM-set A of M, the rb-closure of A is defined as the M-set intersection of all rb-closed M-sets of (M, τ) containing A and is denoted by rb-cl (A). i.e. rb-cl(A) = ⋂ { K ⊆ M: each K is a rb-closed M-set and A ⊆ K } with Crcl(A)(x) = Min {Ck(x): A ⊆ K, each K is a rb-closed M-set }, for all x ∈ X. Definition 3.9: Let (M, τ) be an M-topological space in [X] W. Any subM-set A of M is called an α-open M-set if A ⊆ int (cl (int (A))) with CA(x) ≤ Cint(cl(int(A)))(x), for all x ∈ X. The complement of an α-open M-set is called an α-closed M-set.

Definition 3.10: Let (M, τ) be an M-topological space in [X]W and A be any sub M-set of M. The M-set intersection of all α-closed M-sets containing A is called the α-closure of A and is denoted by αcl (A). Similarly, the M-set union of all α- open M-sets of (M, τ) contained in A is called the α-interior of A and is denoted by αint (A). i.e. αcl(A) = ⋂ { K ⊆ M: each K is an α-closed M-set and A ⊆ K } with Cαcl(A)(x) = Min {Ck(x): A ⊆ K, each K is an α-closed M-set }, for all x ∈ X. Similarly, αint (A) = ⋃ { K ⊆ M: each K is an α-open M- set and A ⊇ K } with Cαint(A)(x) = Max {Ck(x): A ⊇ K, each K is an α-open M-set }, for all x ∈ X. 1/c } }, the collection of the b-open M-sets of (M, τ) is { ϕ, M, { 1/a }, { 2/b }, { 1/b }, { 1/a, 2/b }, { 2/b, 1/c }, { 1/a, 1/c }, { 1/b, 1/a }, { 1/a, 1/b, 1/c } } and the collection of the α- open M-sets of (M, τ) is { M, ϕ, { 2/a }, { 1/b }, { 2/a, 1/b } }.

Let A = {1/a, 1/b, 1/c} be a sub M-set of M and A ⊂ U = M is the α-open M-set in (M, τ). Then αcl (A) = M = U. Therefore, αcl (A) = U with Cαcl (A) (x) = CU(x) whenever A ⊂ U and U is α-open M-set with CA(x) < CU(x), for all x ∈ X. Hence, A is a gα-closed M-set.

But A = U = {1/a, 1/b, 1/c} is the b-open M- set. Then rcl (A) = M ⊈ U with Crcl (A) (x) ≰ CU(x), for all x ∈ X, whenever A = U and U is b-open M-set. Hence, A is not a rb-closed M-set. Remark 3.1.10: From the above discussions, the following implications hold: In the following diagram by A B means A implies B A B does not imply A. 3.2 Regular b-continuous M-set functions Definition 3.2.1 Let (M, τ) and (N, 𝜎) be any two M- topological spaces.

Any M-set function f: (M, τ) ⟶ (N, 𝜎) is said to be a continuous M-set function if f ⎼1 (A) is closed (resp. open) M-set in (M, τ), for every closed (resp. open) M- set A in (N, 𝜎). Definition 3.2.2 Let (M, τ) and (N, 𝜎) be any two M- topological spaces. Any M-set function f: (M, τ) ⟶ (N, 𝜎) is said to be a regular continuous M-set function if f ⎼1 (A) is regular closed (resp. regular open) M-set in (M, τ), for every closed (resp. open) M-set A in (N, 𝜎). Definition 3.2.3 Let (M, τ) and (N, 𝜎) be any two M- topological spaces.

Any M-set function f: (M, τ) ⟶ (N, 𝜎) is said to be a regular b-continuous (rb-continuous) M-set function if f ⎼1 (A) is rb-closed (res. rb-open) M-set in (M, τ), for every closed (res. open) M-set A in (N, 𝜎). Definition 3.2.4 Let (M, τ) and (N, 𝜎) be any two M- topological spaces. Any M-set function f: (M, τ) ⟶ (N, 𝜎) is said to be a generalized α-continuous (gα-continuous) M-set function if f ⎼1 (A) is gα-closed (res. gα-open) M-set in (M, τ), for every closed (res. open) M-set A in (N, 𝜎). Proposition 3.2.5 Let (M, τ) and (N, 𝜎) be any two M- topological spaces in [X] W.

For any M-set function f: (M, τ) → (N, 𝜎), the following statements are equivalent: i. f is a rb-continuous M-set function. ii. f(rb-cl(A)) ⊆ cl(f(A)), for each A ⊆ M with Cf(rb-cl(A))(x) ≤ Cclf(A) (x), for all x ∈ X iii. rb-cl((B)) ⊆ f −1(cl(B)), for each B ⊆ N with Crb-cl(f ⎼1(B))(x) ≤ Cf ⎼1(cl(B))(x), for all x ∈ X. iv. f −1(int(B)) ⊆ rb-int(f −1(B)), for each B ⊆ N with v. C (f ⎼1(int(B))(x) ≤ Crb-int(f ⎼1 (B))(x), for all x ∈ X. Proposition 3.2.6: Every regular continuous M-set function is a continuous M-set function. Remark 3.2.7: The converse of the above Proposition need not be true as shown in the Example 2.2.5 Example 3.2.8: Let X = {a, b, c}, W1 = 2, Y = {x, y, z}, and W2 = 1.

Let M = {1/a, 2/b, 1/c} and N = {1/x, 1/y, 1/z} be two M-sets. Let τ = { M, ϕ, { 1/a }, { 2/b }, { 1/a, 2/b } } and 𝜎 = { N, ϕ, { 1/x, 1/y } } be the M-topologies on M and N respectively. Then (M, τ) and (N, 𝜎) are M-topological spaces. Now the collection of the closed M-sets of (N, 𝜎) is { ϕ, N, { 1/z } } and the collection of the regular closed M-sets of (M, τ) is { M, ϕ, { 2/b, 1/c }, { 1/a, 1/c } } and the collection of the closed M-sets of (M, τ) is { ϕ, M, { 2/b, 1/c}, {1/a, 1/c}, {1/c}}.

Let M-set function f: (M, τ) ⟶ (N, 𝜎) be defined by f = { (1/a, 1/x)/1, (2/b, 1/y)/2, (1/c, 1/z)/1 }. For the closed M-set A = {1/z} in (N, 𝜎), f −1 (A) = {1/c} is an closed M-set in (M, τ). Trivially f ⎼1 (ϕ) = ϕ and f ⎼1(N) = M are closed M-sets in (M, τ). But {1/c} is not a regular closed M-set in (M, τ).

Hence, every continuous M-set function need not be a regular continuous M-set function. Proposition 3.2.9: Every regular continuous M-set function is a rb-continuous M-set function. Remark 3.2.10: The converse of the Proposition 2.2.3 need not be true as shown in following Example. Example 3.2.11: Let X = {a, b, c}, W1 = 2, Y = {x, y, z}, and W2 = 1.

Let M = {1/a, 2/b, 1/c} and N = {1/x, 1/y, 2/z} be two M-sets. Let τ = { M, ϕ, { 1/a }, { 2/b }, { 1/a, 2/b } } and 𝜎 = { N, ϕ, { 1/x, 1/y } } be two M-topologies on M and N respectively. Then (M, τ) and (N, 𝜎) are M-topological spaces. Now the collection of the regular closed M-sets of (M, τ) is { M, ϕ, { 2/b, 1/c }, { 1/a, 1/c } } and the collection of the rb-closed M-sets of (M, τ) is { M, ϕ, { 1/c, 2/b }, { 1/a, 1/c }, { 1/c } } and the collection of the closed M-sets of (N, 𝜎) is { ϕ, N, { 2/z } }.

Let M-set function f: (M, τ) ⟶ (N, 𝜎) be defined by f = { (1/a, 1/x)/1, (2/b, 1/y)/2, (1/c, 2/z)/2 }. For the closed M-set A = {2/z} in (N, 𝜎), f −1 (A) = {1/c} is an rb-closed M-set in (M, τ). Trivially f ⎼1 (ϕ) = ϕ and f ⎼1 (N) = M are rb-closed M-sets in (M, τ). But {1/c} is not a regular closed M-set in (M, τ).

Hence, every rb-continuous M-set function need not be a regular continuous M-set function. Proposition 3.2.12: Every rb-continuous M-set function is a gα-continuous M-set function. Remark 3.2.13: The converse of the Proposition 2.2.4 need not be true as shown in the Example 2.2.7 Example 3.2.14: Let X = {a, b, c}, W1 = 2, Y = {x, y, z} and W2 = 2. Let M = {1/a, 2/b, 1/c} and N = {2/x, 1/y, 1/z} be two M-sets.

Let τ = { ϕ, M, { 1/a }, { 2/b }, { 1/a, 2/b } } and 𝜎 = { N, ϕ, {2/x}} be the M-topologies on M and N respectively. Then (M, τ) and (N, 𝜎) are M-topological spaces. Now the collection of the closed M-sets of (N, 𝜎) is { ϕ, N, { 1/y, 1/z } } and the collection of the rb-closed M-sets of (M, τ) is { M, ϕ, { 1/c }, { 2/b, 1/c }, { 1/c, 1/a } } and the collection of the gα-closed M-sets of (M, τ) is { M, ϕ, { 1/c }, { 1/b, 1/c }, { 1/c, 2/b }, { 1/c, 1/a }, { 1/a, 1/b, 1/c } }. Let M-set function f: (M, τ) ⟶ (N, 𝜎) be defined by f = { (1/a, 2/x)/2, (1/b, 1/y)/2, (1/c, 1/z)/1 }.

For the closed M-set A = {1/y, 1/z} in (N, 𝜎), f −1 (A) = {1/b, 1/c} is a gα-closed M- set in (M, τ). Trivially f ⎼1 (ϕ) = ϕ and f ⎼1 (N) = M are gα- all x ∈ X. Such a subcollection B satisfies the following conditions: 1. ϕ ∉ B 2. For every B1, B2 ∈ B there exists B3 ∈ B with B3 ⊆ B1 ⋂ B2 with CB3 (x) ≤ CB1 (x) ⋂ CB2 (x), for all x ∈ X.

Definition 4.3: Let F be a filter on B. Then F is a maximal filter if every filter including F coincides with F Definition 4.4: Let (M, τ) be an M-topological space. A filter base B = { { Aα }α 𝜺 J, where J is an indexed set and Aα ∈ M } rb-converges to a point xo ∈m M if for each rb-open M-set V containing m/xo there exists an Aα ∈ B such that Aα ⊆ rb-cl(V) with CAα (x) ≤ Crb-cl(V) (x), for all x ∈ X. Definition 4.5: A filter base B = { { Aα }α ∈ J, where J is indexed set and Aα ∈ M } rb-accumulates to a point xo ∈m M if for each rb-open M-set V containing m/xo and Aα ∈ B, Aα ⋂ rbcl (V) ≠ ϕ with Min { CAα(x), Crb-cl (V) (x) } ≠ 0, for all x ∈ X.

Definition 4.6: Let (M, τ) be an M-topological space. A collection 𝒜 of sub M-sets of a space M is said to cover M, or to be a covering of M, if the union of the elements of 𝒜 is equal to M. It is called an open covering of M if its elements are open sub M-sets of M. Definition 4.7: Let (M, τ) be an M-topological space.An M-topologic closed M-sets in (M, τ), but {1/b, 1/c} is not a rb-closed in (M, τ).

Hence every gα-continuous M-set function need not be a rb-continuous M-set function. Proposition 3.2.15: Every regular continuous M-set function is gα-continuous M-set function. Remark 3.2.16: The converse of the Proposition 2.2.5 need not be true as shown in the Example 2.2.6 Example 3.2.17: Let X = {a, b, c}, W1 = 2, Y = {x, y, z} and W2 = 2. Let M = {1/a, 2/b, 1/c} and N = {2/x, 1/y, 1/z} be two M-sets.

Let τ = { ϕ, M, { 1/a }, { 2/b }, { 1/a, 2/b } } and 𝜎 = { N, ϕ, { 2/x, 1/y } } be the M-topologies on M and N respectively. Then (M, τ) and (N, 𝜎) are M-topological spaces. Now the collection of the closed M-sets of (N, 𝜎) is { ϕ, N, { 1/z } } and the collection of the regular closed M-sets of (M, τ) is { M, ϕ, { 2/b, 1/c }, { 1/c, 1/a } } and the collection of the gα-closed M-sets of (M, τ) is { M, ϕ, { 2/b, 1/c }, { 1/c }, { 1/b, 1/c }, { 1/c, 1/a }, { 1/a, 1/b, 1/c } }. Let M-set function f: (M, τ) ⟶ (N, 𝜎) be defined by f = { (1/a, 2/x)/2, (2/b, 1/y)/2, (1/c, 1/z)/1 }.

For the closed M-set A = {1/z} in (N, 𝜎), f −1 (A) = {1/c} is an gα-closed M-set in (M, τ). Trivially f ⎼1 (ϕ) = ϕ and f ⎼1 (N) = M are gα-closed M-sets in (M, τ), but {1/c} is not a regular closed in (M, τ). Hence, every gα-continuous M-set function need not be a regular continuous M-set function. Remark 3.2.18: From the above discussions, the following implications hold: In the following diagram by A B means A implies B A B does not imply A. -S-closed if for every cover { Vα\α ∈ I, I is an indexed set } of (M, τ) by rb-open M- sets of M, there exists a finite subM-set I0 of I such that ⋃ { rb-cl (Vα)\α ∈ I0 } = M with Max { Crb-cl(Vα) (x)\ α ∈ I0 } = CM (x), for all x ∈ X.

Definition 4.8: Let (M, τ) be an M-topological space. A subM-set K of an M-topological space (M, ) is said to be rb- closed relative to M if for any cover {Ui: i ∈ I, I is an indexed set } of K by rb-open M-sets, there exists a finite subM-set I0 of I such that K ⊆ ⋃ { rb-cl (Ui)\ i ∈ I0 } with CK (x) ≤ Max { Crb-cl(Ui)(x)\ i ∈ I0 }, for all x ∈ X. Remark 4.9: The family of all rb-open (res, rb-closed) M-sets of (M, τ) is denoted by RBO(X) (res, RBC(X)). The family of all rb-open M-sets of (M, τ) containing a point x ∈m M is denoted by RBO (M, m/x).

Remark 4.10: A filter base B is rb-convergent to a point x ∈m M if and only if B contains the collection {rbcl (U): U ∈ RBO (M, m/x)}. Proposition 4.11: For an M-topological space (M, τ), the following statements are equivalent: 1. M is an rb-S-closed M-space. 2. For each family of rb-closed M-sets { Fα }α∈I, where I is an indexed set, such that ⋂ (Fα) = ϕ with Min { CFα (x), α ∈ I } = 0, for all x ∈ X, there exists a finite subfamily n 4.

On rb-Convergence in Multiset Topological Spaces In this chapter, the concepts of rb-convergence to a point, rb- accumulates to a point, rb-S-closed M-spaces are studied and some interesting properties are discussed. Finally, sequence of rb-convergence and rb-Hausdorff M-spaces are discussed. Definition 4.1: A filter is a non-empty collection F of subM- sets of an M- topological space (M, τ) such that 1. ϕ ∉ F; 2. If A ∈ F and B ⊇ A with CB (x) ≥ CA(x), for all x ∈ X, 𝑛 such that  {𝐹∝𝑖}𝑖=1 1 i  rb-int (Fαi) = ϕ with Min { Crb- then B ∈ F and 3.

If A ∈ F and B ∈ F, then A ⋂ B ∈ F with CA ⋂ B (x) ≤ CF int(Fαi)(x), i = 1, 2,…n } = 0, for all x ∈ X. 3. Each filter base B = { Aα }α ∈ I with CB (x) = C{ Aα }α ∈ I(x), for all x ∈ X, rb-accumulates to some point xo ∈m M. 4. Each maximum filter base B Rb-converges. (x), for all x ∈ X. Definition 4.2: Let (M, τ) be an M-topological space.

A filter base of a filter F is a sub collection B of M such that for every F ∈ F there exists B ∈ B with B ⊆ F with CB (x) ≤ CF (x), for 9. Maki H, Devi R, Balachandran K. Associated topologies of generalized α-closed sets and α-generalized closed sets, Mem, Fac. Sci, Kochi Univ.

Ser. A Math. 1994; 15:51-63. 10. Nagaveni N, Narmadha A. A note on regular b-open sets in topological spaces, Proc, UGC and DST-Curie sponsored International Conference on Mathematics and its Applications-A New Wave, ICMANW, Dec. 2011, 93-100, 21-22. 11.

Nagaveni N, Narmadha A. On regular b-closed sets in topological spaces, Heber International Conference on Applications of Mathematics and Statistics, HICAMS, 2012, 5-7, 81-87. 12. Narmatha A, Nagaveni N, Noiri T. On Regular b-open sets in Topological spaces, Int.

Jornal of Math. Analysis. 2013; 7(19):937-948. 13. Stone MH. Applications of the theory of Boolean rings to general topology”, Trans.

Math, Soc. 1937; 41:375-381. 14. Thamson T. S-closed spaces, Proc. Amer.

Math, Soc. 1976; 60:335-338. 15. Thomas Wright. Convergence of sequences and Nets in Metric and Topological space, July 1, 2013, 1-6. Proposition 4.12: A subM-set A of a M-topological space (M, τ) is rb-closed relative to M if and only if every filter base B on M with A ∈ B rb-converges to a point in A.

Definition 4.13: Let (M, τ) be an M-topological space. A sequence in a M-set M is a function f: ℕ → M. Then m/xn = f(n) for all n ∈ ℕ, where ℕ is positive integer and denoted the sequence as (m/xn)n ∈ ℕ. Definition 4.14: Let (M, τ) be an M-topological space, let x ∈m M and N ⊆ M.

Then N is said to be a rb-neigh bor hood of m/x if there is an rb-open M-set V in τ such that x ∈m V and CV(y) ≤ CN(y), for all y x. I.e. a rb-neighborhood of m/x in M means any rb-open M-set containing m/x. Here m/x is said to be an interior point of N. Definition 4.15: Let (M, τ) be an M-topological space.

A sequence m/x1, m/x2, m/x3… of points in M is said to be rb- converges to a point x ∈m M if for each rb-neighborhood U of m/x, there is a positive integer ℕ such that xn ∈m U, for all n ≥ ℕ. It is denoted by (m/ xn) → m/x. Definition 4.16: An M-topological space (M, τ) is said to be a rb-Hausdorff M-space, if each pair of distinct points m/x and m/y in M, there exist rb-open M-sets U and V containing m/x and m/y respectively such that U ⋂ V = ϕ with CU ⋂ V(x) = 0, for all x ∈ X. Definition 4.17: Let (M, τ) be an M-topological space.

A rb- countable basis at a point m/x is a countable M-set { Un | n ∈ ℕ } of rb-neighbourhood of m/x, such that for any rb- neighbourhood V of m/x there is an n ∈ ℕ such that Un ⊆ V with CUn(x) ≤ CV(x), for all x ∈ X. An M-topological space (M, τ) is said to be first rb-countable if every point has a rb-countable basis. Proposition 4.3 Let (M, τ) be an M-topological space. Let M be a rb-Hausdorff space and xn ∈m M a rb-convergent sequence.

Then the limit lim n→∞m/xn is unique.

Acknowledgement

The authors would like to express their sincere gratitude to the referees for their valuable suggestions and comments for the betterment of this paper.

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