Generalized Closed Sets Via Grills.pdf

(a) A ⊆B ⇒Φ(A) ⊆Φ(B).
(b) Φ(A S B) = Φ(A) S Φ(B).
(c) Φ(Φ(A)) ⊆Φ(A) = cl(Φ(A)) ⊆cl(A), and hence Φ(A) is closed in (X, τ), for
all A ⊆X.
Theorem 1.4 [9] Let (X, τ) be a topological space and G be a grill on X. Then
β(G, τ) = {V \A: V ∈τ and A /∈G } is an open base for τG.
2. Generalized closed sets with respect to a grill
We begin by introducing a new class of generalized closed sets in terms of grills
as follows.
Definition 2.1 Let (X, τ) be a topological space and G be a grill on X. Then a
subset A of X is said to be g-closed with respect to the grill G (G-g-closed, for short)
if Φ(A) ⊆U whenever A ⊆U and U is open in X.
A subset A of X is said to be G-g-open if X \ A is G-g-closed.
Remark 2.2 For a topological space (X, τ) and a grill G on X, we obtain as
follows.
(a) Every closed set in X is G-g-closed.
(b) As Φ(Φ(A)) ⊆Φ(A), Φ(A) is G-g-closed for any subset A of X, and hence
every τG-closed set is G-g-closed.
(c) It is known [9] that for any A /∈G, Φ(A) = φ. Thus any non-member of G is
G-g-closed.
(d) Every g-closed set is G-g-closed; that the converse is false is shown by the
following example.
Example 2.3 Let X={a,b,c}, τ={φ,{b},{b,c}, X} and G = {{a},{c},{a,c},{a,b},
{b,c},X}.
Then (X, τ) is a topological space and G is a grill on X.
Suppose
A = {b}. Then it is easy to verify that A is not g-closed but is G-g-closed. In fact,
A ⊆{b,c} but cl(A)=X ̸⊆{b,c}, and Φ(A) = φ.
Corresponding to any nonempty subset A of X, a typical grill [A] on X was
defined in [10] in the following manner.
Definition 2.4 Let X be a space and (φ ̸=)A ⊆X. Then
[A] = {B ⊆X : A T B ̸= φ}
is a grill on X, called the principal grill generated by A.
Remark 2.5 In the case of principal grill [X] generated by X, it is known [10] that
τ = τ[X], so that any [X]-g-closed set becomes simply a g-closed set and vice-versa.
Next we observe that in a space X, Gδ = {A ⊆X : int(cl(A)) ̸= φ} is a grill
on X, and for this grill we have:
Theorem 2.6 Let (X, τ) be a topological space and A ⊆X. Then τGδ = τ α and
hence a subset A of X is Gδ-g-closed iffA is αg-closed.
Proof: It is well known that for any subset A of a space (X, τ), α-clA = A S
cl(int(cl(A))). Now, with the grill Gδ we have for any A ⊆X,
Φδ(A)= {x ∈X : U T A ∈Gδ, ∀U ∈τ(x)} = {x ∈X : int(cl(U T A)) ̸= φ, ∀
U ∈τ(x)} = cl(int(cl(A))). Thus τGδ-cl(A) = A S Φ(A) = A S cl(int(cl(A))) =
α-cl(A). Hence τGδ = τ α.
Theorem 2.7 Let (X, τ) be a topological space and G be a grill on X. Then for a
subset A of X, the following are equivalent:
(a) A is G-g-closed.
(b) τG-cl(A)⊆U whenever A ⊆U and U is open.
(c) For all x ∈τG-cl(A), cl({x})T A ̸= φ.
(d) τG-cl(A)\A contains no nonempty closed set of (X, τ).
(e) Φ(A)\A contains no nonempty closed set of (X, τ).
Proof: (a)⇒(b): Suppose A is G-g-closed and A ⊆U where U is open in (X, τ).
Then Φ(A) ⊆U so that τG-cl(A)= A S Φ(A) ⊆U.
(b)⇒(c): Suppose x ∈τG-cl(A). If cl({x}) T A = φ, then A ⊆X\ cl({x}) and
using (b), τG-cl(A) ⊆X\ cl({x}), a contradiction since x ∈τG-cl(A).
(c)⇒(d): Suppose F is a closed set of (X, τ) contained in τG-cl(A) \ A and x ∈F.
Since F T A = φ, we have cl({x}) T A = φ. Again since x ∈τG-cl(A), by (c) we
have cl({x}) T A ̸= φ, a contradiction. This proves (d).
(d)⇒(e): It follows from the fact that τG-cl(A) \A = Φ(A) \A.
(e)⇒(a): Suppose that A ⊆U and U is open in (X, τ). Since Φ(A) is closed
(by Th