Topology on grill-filter space and continuity

Bol. Soc. Paran. Mat. c SPM –ISSN-2175-1188 on line SPM: www.spm.uem.br/bspm (3s.) v. 31 2 (2013): 219–230. ISSN-00378712 in press doi:10.5269/bspm.v31i2.16603 Topology of Grill Filter Space and Continuity Shyamapada Modak abstract: This paper will discuss about a new topology, obtained from a grill and a filter on the same set. The Characterizations and open base of the new topology are also aim of this paper. The generalized continuity is also a part of this paper. Key Words: grill-filter space, Ω-operator, ψ Ω -operator, τ ψ FG -topology, F-continuity. Contents 1 Introduction 219 2 Preliminaries 220 3 ψ Ω -C set 223 4 τψ FG -topology 224 5 Continuity on grill-filter topological spaces 228 1. Introduction The notion of grill [7] and filter [21] is already in literature from 1947 and 1937 respectively. The topics - Proximity spaces, Closure spaces, the Theory of Compactifications and similar other extension problems [5,6,7,20] have been enriched by the study of grill. The filter is an important part in topological space for the discussion of the separation axioms, compactness, continuity etc. Recently mathematicians: Roy and Mukherjee [19], Noiri and Al-Omiri [1,2,3] have used grill on topological space as like ideal topological space [4,8,10,11,17,22] and have obtained many new topologies. Further Noiri and Al-Omiri and Modak et al [13,14,15,16] have considered ideal or grill on generalized spaces and discussed different types of topological space. In this paper, we shall use grill and filter in different aspect something different from traditional uses of the same. Actually we shall define a space with grill and filter together on a set. From this space we define a topology via two operators. We also give a standard form of base, and characterize the topology. We also discuss a new type of generalized continuity on the new topological space. At last we shall obtain the relations of this continuity with usual continuity. 2000 Mathematics Subject Classification: 54A05, 54C10 219 Typeset by BSP style. M c Soc. Paran. de Mat. 220 Shyamapada Modak 2. Preliminaries In this section we shall give some definitions and prove some results, which are the preliminaries for the paper. At first we shall give the formal definition of filter. A subcollection F (not containing the empty set) of ℘(X) is called a filter [21] on X if F satisfies the following conditions: 1. A ∈ F and A ⊆ B implies B ∈ F; 2. A, B ∈ F implies A ∩ B ∈ F. In this paper we shall try to obtain a topology with the help of filter, for this we shall discuss following: Definition 2.1. A set A ∈ ℘(X) is called an F-open set if A ∈ F. B ∈ ℘(X) is called a F-closed set if X \ B ∈ F. We set F-Int(A) = ∪{U : U ⊆ A, U ∈ F} and F-Cl(A) = ∩{F : A ⊆ F, X \ F ∈ F}. Here we shall prove some theorems related to F-Int and F-Cl: Theorem 2.1. Let F be a filter on X and A ⊆ X. Then x ∈ F-Cl(A) if and only if every F-open set Ux containing x, Ux ∩ A 6= φ. Proof: Let x ∈ F-Cl(A). Supposed that Ux ∩ A = φ, where Ux is an F-open set containing x. Then A ⊆ (X \ Ux ) and X \ Ux is a F-closed set containing A. Therefore x ∈ / (X \ Ux ), and this is a contradiction. Conversely supposed that Ux ∩ A 6= φ, for every F-open set Ux containing x. If possible suppose that x ∈ / FCl(A), then there exists F subset of X which satisfy A ⊆ F, X \ F ∈ F and x ∈ / F. Therefore x ∈ (X \ F ). So A ∩ (X \ F ) = φ for an F-open set X \ F containing x. It is a contradiction. ✷ Theorem 2.2. Let F be a filter on X and A ⊆ X. Then F-Int(A) = X \ FCl(X \ A). Proof: Let x ∈ F-Int(A). Then there is an U ∈ F, such that x ∈ U ⊆ A. Hence x∈ / (X \ U ), i.e., x ∈ / F-Cl(X \ U ), since X \ U is a F-closed set containing X \ A. So x ∈ / F-Cl(X \ A)(from Definition 2.1), and hence x ∈ X \ F-Cl(X \ A). Conversely suppose that x ∈ X \ F-Cl(X \ A). So x ∈ / F-Cl(X \ A), then there is an F-open set Ux containing x, such that Ux ∩ (X \ A) = φ. So Ux ⊆ A. Therefore x ∈ F-Int(A)(from Definition 2.1). Hence the result. ✷ Theorem 2.3. Let F be a filter on X and A ⊆ X. Then for G ∈ F, G ∩ FCl(A) ⊆ F-Cl(G ∩ A). Proof: Let x ∈ G ∩ F-Cl(A). Then x ∈ G and x ∈ F-Cl(A). Implies that x ∈ G and for every F-open set Ux containing x, Ux ∩ A 6= φ. Again G ∩ Ux is an F-open Topology of Grill Filter Space and Continuity 221 set containing x, then (G ∩ Ux ) ∩ A 6= φ. Hence x ∈ F-Cl(G ∩ A). Therefore G ∩ F-Cl(A) ⊆ F-Cl(G ∩ A). ✷ Following is the concepts of grill [7]: A subcollection G (not containing the empty set) of ℘(X) is called a grill [7] on X if G satisfies the following conditions: 1. A ∈ G and A ⊆ B implies B ∈ G; 2. A, B ⊆ X and A ∪ B ∈ G implies that A ∈ G or B ∈ G. Let F and G be the filter and grill respectively on the same set X. Then (X, F, G) is denoted as grill-filter space. One of the operator on grill-filter space is: Definition 2.2. [14]. Let (X, F, G) be a grill-filter space. A mapping ΩG : ℘(X) → ℘(X) is defined as follows ΩG (A) = Ω(A) = {x ∈ X : A ∩ U ∈ G for all U ∈ F(x)} for each A ∈ ℘(X), where F(x) = {U ∈ F : x ∈ U }. Here we shall mention a property on Ω-operator, although so many properties have been discussed in [14]. Theorem 2.4. Let (X, F, G) be a grill-filter space and A, B ⊆ X. Then Ω(A∩B) ⊆ Ω(A) ∩ Ω(B). Proof: Since A∩B ⊆ A and A∩B ⊆ B, then Ω(A∩B) ⊆ Ω(A) [14] and Ω(A∩B) ⊆ Ω(B) [14]. Hence Ω(A ∩ B) ⊆ Ω(A) ∩ Ω(B). ✷ Following example shows that the reverse inclusion of the above theorem does not hold in general, however the relation, Ω(A ∪ B) = Ω(A) ∪ Ω(B) [14] hold. Example 2.1. Let X = {a, b, c, d}, F = {X, {a, b, c}} and G = {{a}, {b}, {a, c}, {a, b}, {a, d}, {a, b, c}, {c, b, d}, {a, b, d}, {a, c, d}, {b, c}, {b, d}, X}. Consider A = {a, d}, and B = {b, d}. Then Ω({a, d}) = {a, b, c, d} and Ω({b, d}) = {a, b, c, d}, and hence Ω(A) ∩ Ω(B) = {a, b, c, d}. But Ω(A ∩ B) = Ω({d}) = φ. So, Ω(A) ∩ Ω(B) is not a subset of Ω(A ∩ B). New topology from grill-filter space is: Remark 2.1. [14]. Let (X, F, G) be a grill-filter space. We define a map CL : ℘(X) → ℘(X) by CL(A) = A ∪ Ω(A), for all A ∈ ℘(X). Then the map ′ CL′ is a Kuratowski closure operator. We will denote τ FG , the topology, generated by CL, that is τ FG = {V ⊆ X : CL(X \ V ) = X \ V }. In this paper we shall denote interior and closure operator of (X, τ FG ) by IntFG and ClFG respectively. Again (X, τ FG , F, G) will be denoted as grill-filter topological space. Following is the representation of an open base for the topology τ FG : 222 Shyamapada Modak Theorem 2.5. [14]. Let (X, F, G) be a grill-filter space. Then β(F, G) = {V \G: V ∈ F, G∈ / G} is an open base for the topology τ FG . Another operator on (X, F, G) is defined as follows: Definition 2.3. [14]. Let (X, F, G) be a grill-filter space. An operator ψ Ω : ℘(X) → F is defined as follows for every A ∈ ℘(X), ψ Ω (A) = {x ∈ X: there exists U ∈ F(x) such that U \ A ∈ / G} and observe that ψ Ω (A) = X \ Ω(X \ A). Now we shall prove some characterizations: Theorem 2.6. Let (X, F, G) be a grill-filter space. Then F ⊆ (...truncated)