Download On Ψ~ e G-sets in grill topological spaces

Faculty of Sciences and Mathematics, University of Niš, Serbia
Available at: http://www.pmf.ni.ac.rs/filomat
Filomat 25:2 (2011), 187–196
DOI: 10.2298/FIL1102187A
e G -SETS IN GRILL TOPOLOGICAL SPACES
ON Ψ
Ahmad Al-Omari and Takashi Noiri
Abstract
e G -sets and utilize the ΨG -operator
In this paper, we introduce and study Ψ
to define interesting generalized open sets and study their properties.
1
Introduction
The idea of grills on a topological space was first introduced by Choquet [4]. The
concept of grills has shown to be a powerful supporting and useful tool like nets and
filters, for getting a deeper insight into further studying some topological notions
such as proximity spaces, closure spaces and the theory of compactifications and
extension problems of different kinds (see [3], [2], [18] for details). In [15], Roy
and Mukherjee defined and studied a typical topology associated rather naturally
to the existing topology and a grill on a given topological space.
The notion of ideal topological spaces was first studied by Kuratowski [8] and
Vaidyanathaswamy [19]. Compatibility of the topology τ with an ideal I was
first defined by Njåstad [13]. In 1990, Jankovic and Hamlett [7, 5] investigated
further properties of ideal topological spaces and another operator called Ψ-operator
defined as Ψ(A) = X − (X − A)∗ . In 2007 Modak and Bandyopadhyay [11] used
the Ψ-operator to define interesting generalized open sets.
2010 Mathematics Subject Classifications. 54A05, 54C10
e G -set, semi-open set
Key words and Phrases. grill, ΨG -operator, α-set, Ψ
Received: December 5, 2010; Revised: February 15, 2011
Communicated by Ljubiša D.R. Kočinac
The authors wishes to thank the referees for useful comments and suggestions.
188
Ahmad Al-Omari and Takashi Noiri
Definition 1.
[17] Let (X, τ, G) be a grill topological space. An operator ΨG :
P(X) → P(X) is defined as follows for every A ∈ X, ΨG (A) = {x ∈ X : there
exists a U ∈ τ (x) such that U − A ∈
/ G} and observes that ΨG (A) = X − Φ(X − A)
(we used the notion ΨG is instead of ΓG which used in [17]).
e G -sets and utilize the ΨG -operator to
In this paper, we introduce and study Ψ
define interesting generalized open sets and study their properties.
2
Preliminaries
Let (X, τ ) be a topological space with no separation properties assumed. For a
subset A of a topological space (X, τ ), Cl(A) and Int(A) denote the closure and
the interior of A in (X, τ ), respectively. The power set of X will be denoted by
P(X). A subcollection G (not containing the empty set) of P(X) is called a grill
[4] on X if G satisfies the following conditions:
1. A ∈ G and A ⊆ B implies that B ∈ G,
2. A, B ⊆ X and A ∪ B ∈ G implies that A ∈ G or B ∈ G.
For any point x of a topological space (X, τ ), τ (x) denotes the collection of all open
neighborhoods of x.
Definition 2.
[15] Let (X, τ ) be a topological space and G be a grill on X. A
mapping Φ : P(X) → P(X) is defined as follows: Φ(A) = ΦG (A, τ ) = {x ∈ X :
A ∩ U ∈ G for all U ∈ τ (x)} for each A ∈ P(X). The mapping Φ is called the
operator associated with the grill G and the topology τ .
Proposition 1. [15] Let (X, τ ) be a topological space and G be a grill on X. Then
for all A, B ⊆ X:
1. A ⊆ B implies that Φ(A) ⊆ Φ(B),
2. Φ(A ∪ B) = Φ(A) ∪ Φ(B),
3. Φ(Φ(A)) ⊆ Φ(A) = Cl(Φ(A)) ⊆ Cl(A).
e G -sets in grill topological spaces
On Ψ
189
Let G be a grill on a space X. Then a mapping Ψ : P(X) → P(X) is defined
by Ψ(A) = A ∪ Φ(A) for all A ∈ P(X) [15]. The map Ψ is a Kuratowski closure
axiom. Corresponding to a grill G on a topological space (X, τ ), there exists a
unique topology τG on X given by τG = {U ⊆ X : Ψ(X − U ) = X − U }, where for
any A ⊆ X, Ψ(A) = A ∪ Φ(A) = τG -Cl(A). For any grill G on a topological space
(X, τ ), τ ⊆ τG . By τG -Int(A), we denote the interior of A with respect to τG . If
(X, τ ) is a topological space with a grill G on X, then we call it a grill topological
space and denote it by (X, τ, G).
Theorem 1. [15] Let (X, τ, G) be a grill topological space. Then β(G, τ ) = {V −G :
V ∈ τ, G ∈
/ G} is an open base for τG .
Corollary 1. [15] Let (X, τ, G) be a grill topological space and suppose A, B ⊆ X
with B ∈
/ G. Then Φ(A ∪ B) = Φ(A) = Φ(A − B).
Several basic facts concerning the behavior of the operator ΨG are included in
the following theorem.
Theorem 2.
[17]
Let (X, τ, G) be a grill topological space. Then the following properties hold:
1. If A, B ∈ P(X), then ΨG (A ∩ B) = ΨG (A) ∩ ΨG (B).
2. If U ∈ τG , then U ⊆ ΨG (U ).
3. If A ∈
/ G, then ΨG (A) = X − Φ(X).
4. If A, B ⊆ X and (A − B) ∪ (B − A) ∈
/ G, then ΨG (A) = ΨG (B).
5. If A ⊆ X, then ΨG (A) is open in (X, τ ).
6. If A ⊆ B, then ΨG (A) ⊆ ΨG (B).
7. If A ⊆ X, then ΨG (A) ⊆ ΨG (ΨG (A)).
8. If A ⊆ X, then A ∩ ΨG (A) = τG -Int(A).
9. If A ⊆ X and G ∈
/ G, then ΨG (A − G) = ΨG (A).
10. If A ⊆ X and G ∈
/ G, then ΨG (A ∪ G) = ΨG (A).
190
3
Ahmad Al-Omari and Takashi Noiri
e G -Sets
Ψ
Definition 3. Let (X, τ, G) be a grill topological space. A subset A of X is called
e G -set if A ⊆ Cl(ΨG (A)).
aΨ
e G -sets in (X, τ, G) is denoted by Ψ
e G (X, τ ).
The collection of all Ψ
e G -sets in a grill
Proposition 2. Let {Aα : α ∈ ∆} be a collection of nonempty Ψ
e G (X, τ ).
topological space (X, τ, G), then ∪α∈∆ Aα ∈ Ψ
Proof. For each α ∈ ∆,
Aα ⊆ Cl(ΨG (Aα )) ⊆ Cl(ΨG (∪α∈∆ Aα ))
This implies that
∪α∈∆ Aα ⊆ Cl(ΨG (∪α∈∆ Aα ))
e G (X, G).
Thus ∪α∈∆ Aα ∈ Ψ
e G -sets in (X, τ, G)
The following example shows that the intersection of two Ψ
e
may not be a ΨG -set.
Example 1.
Let X = {a, b, c, d}, τ = {φ, X, {a}, {b, c}, {a, b, c}} and the grill
G = {{a}, {b}, {a, c}, {a, b}, {a, d}, {a, b, c}, {c, b, d}, {a, b, d}, {a, c, d}, {b, c},
e G -sets but A ∩ B is
{b, d}, {b, c, d}, X}. Then A = {a, d} and B = {b, c, d} are Ψ
e G -set. For A = {a, d}, Φ(X − A) = {b, c, d} and Ψ
e G (A) = {a}. Hence
not a Ψ
e G (A)) implies that A is a Ψ
e G -set. For B = {b, c, d}, Φ(X − B) = {a, d}
A ⊆ Cl(Ψ
e G (B) = {b, c}. Hence B ⊆ Cl(Ψ
e G (B)) implies that B is a Ψ
e G -set. On the
and Ψ
e G (B) = φ. Hence
other hand, since A ∩ B = {d}, Φ(X − (A ∩ B)) = X and Ψ
e G (A ∩ B)) implies that A ∩ B is not a Ψ
e G -set.
A ∩ B * Cl(Ψ
Recall that a subset A of X in a topological space (X, τ ) is called an α-set if
A ⊆ Int(Cl(Int(A))). The collection of α-sets in (X, τ ) is denoted by τ α . Njåstad
e G -sets
[12] has shown that τ α forms a topology. Although the intersection of two Ψ
e G -set but we shall prove that intersection of an α-set with a Ψ
e G -set
need not be a Ψ
e G -set.
is a Ψ
Corollary 2. Let (X, τ, G) be a grill topological space. Then U ⊆ ΨG (U ) for every
open set U ∈ τ .
e G -sets in grill topological spaces
On Ψ
191
Proof. Since τ ⊂ τG , the proof follows easily from Theorem 2.
e G (X, τ ). If U ∈ τ α ,
Theorem 3. Let (X, τ, G) be a grill topological space and A ∈ Ψ
e G (X, τ ).
then U ∩ A ∈ Ψ
Proof. We note that if G is open, for any A ⊆ X, G ∩ Cl(A) ⊆ Cl(G ∩ A). Let
e G (X, G). Then by Theorem 2 and Corollary 2 we have
U ∈ τ α and A ∈ Ψ
U ∩ A ⊆ Int(Cl(Int(U ))) ∩ Cl(ΨG (A))
⊆ Int(Cl(ΨG (U ))) ∩ Cl(ΨG (A))
⊆ Cl[Int(Cl(ΨG (U ))) ∩ ΨG (A)]
= Cl[Int(Cl[ΨG (U ) ∩ ΨG (A)])]
= Cl[ΨG (U ) ∩ ΨG (A)]
= Cl[ΨG (U ∩ A)].
e G (X, τ ).
Hence U ∩ A ∈ Ψ
e G (X, τ ). If U ∈ τ ,
Corollary 3. Let (X, τ, G) be a grill topological space and A ∈ Ψ
e G (X, τ ).
then U ∩ A ∈ Ψ
A set D is called a relatively G-dense in a set A if for every relatively nonempty
open set U ∩ A, U ∈ τ , it is true that (U ∩ A) ∩ D ∈ G. We now provide a necessary
e G (X, τ ).
and sufficient condition for A ∈
/Ψ
e G (X, τ ) if and only if there exists x ∈ A
Theorem 4. A set A dose not belong to Ψ
such that there is a neighborhood Vx ∈ τ of x for which X − A is relatively G-dense
in Vx .
e G (X, τ ). We are to show that there exists an element x ∈ A and a
Proof. Let A ∈
/Ψ
neighborhood Vx ∈ τ (x) satisfying that (X − A) is relatively G-dense in Vx . Since
A * Cl(ΨG (A)), there exists x ∈ X such that x ∈ A but x ∈
/ Cl(ΨG (A)). Hence
there exists a neighborhood Vx ∈ τ (x) such that Vx ∩ ΨG (A) = φ. This implies that
Vx ∩ (X − Φ(X − A)) = φ and hence Vx ⊆ Φ(X − A). Let U be any nonempty
open set in Vx . Since Vx ⊆ Φ(X − A), therefore U ∩ (X − A) ∈ G. This implies
that (X − A) is relatively G-dense in Vx . Converse part follows by reversing the
argument.
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Ahmad Al-Omari and Takashi Noiri
Recall that a subset A ⊆ X is said to be preopen [10] (resp. Φ-open [6]) if
A ⊆ Int(Cl(A)) (resp. A ⊆ Int(Φ(A))). The collection of all peropen (resp. Φopen) sets in a topological space (X, τ ) is denoted by P O(X, τ ) (resp. ΦO(X, τ )).
Definition 4. Let (X, τ, G) be a grill topological space. A grill G is said to be
anti-codense grill if τ − {φ} ⊆ G.
Theorem 5. [16] Let (X, τ, G) be a grill topological space, where G is an anticodense grill. Then ΦO(X, τ ) = P O(X, τG ).
Recall that a subset A of X in a topological space (X, τ ) is called a semi-open
set [9] if A ⊆ Cl(Int(A)). The collection of semi-open sets in (X, τ ) is denoted by
e G (X, τ ).
SO(X, τ ). We prove SO(X, τG ) = Ψ
Theorem 6. Let (X, τ, G) be a grill topological space, where G is an anti-codense
e G (X, τ ).
grill. Then SO(X, τG ) = Ψ
Proof. Let A ∈ SO(X, τG ). Therefore by Theorem 2, A ⊆ τG -Cl(τG -Int(A)) = τG e G (X, τ ). Therefore,
Cl(ΨG (A)∩A) ⊆ Cl(ΨG (A)∩A) ⊆ Cl(ΨG (A)) and hence A ∈ Ψ
e G (X, τ ).
SO(X, τG ) ⊆ Ψ
e G (X, τ ) and x ∈ A. Consider a basic neighbourhood U1 of
Conversely, let A ∈ Ψ
x in (X, τG ). Then U1 is of the form U −G where U ∈ τ and G ∈
/ G. This implies that
x ∈ U . Since A ⊆ Cl(ΨG (A)) and U ∈ τ (x), U ∩ ΨG (A) 6= φ. Let y ∈ U ∩ ΨG (A).
Therefore there exists a neighbourhood Wy of y such that Wy −A ∈
/ G (by definition
of ΨG (A)). Now we consider U ∩ Wy = V , let G1 = V − A ∈
/ G (by heredity), then
V 6= φ, V ∈ τ and V − G1 ⊆ A. Also V ⊆ U . Thus M = V − (G1 ∪ G) ⊆ A (note
that M = V − (G1 ∪ G) 6= φ since G is an anti-codense grill) and M ⊆ A ∩ (U − G).
Hence A contains a nonempty τG -open set M contained in U − G. Since x is an
arbitrary point of A, we get A ⊆ τG -Cl(τG -Int(A)) and hence A ∈ SO(X, τG ).
e G (X, τ ) ⊆ SO(X, τG ). Hence SO(X, τG ) = Ψ
e G (X, τ ).
Therefore, Ψ
Definition 5. Let (X, τ, G) be a grill topological space and A ⊆ X, A is called a
ΨA -set if A ⊆ Int(Cl(ΨG (A))).
The collection of all ΨA -sets in (X, τ, G) is denoted by τ A . From Definitions
e G (X, τ ). We show that the collection τ A forms a
3 and 5 it follows that τ A ⊆ Ψ
topology.
e G -sets in grill topological spaces
On Ψ
193
Theorem 7. Let (X, τ, G) be a grill topological space, where G is an anti-codense
grill. Then the collection τ A = {A ⊆ X : A ⊆ Int(Cl(ΨG (A)))} forms a topology
on X.
Proof. (1) It is observed that φ ⊆ Int(Cl(ΨG (φ))) and X ⊆ Int(Cl(ΨG (X))), and
thus φ and X ∈ τ A .
(2) Let {Aα : α ∈ ∆} ⊆ τ A , then ΨG (Aα ) ⊆ ΨG (∪Aα ) for every α ∈ ∆. Thus
Aα ⊆ Int(Cl(ΨG (Aα ))) ⊆ Int(Cl(ΨG (∪Aα ))) for every α ∈ ∆, which implies that
∪Aα ⊆ Int(Cl(ΨG (∪Aα ))). Therefore, ∪Aα ∈ τ A .
(3) Let A, B ∈ τ A . Since ΨG (A) is open in (X, τ ), by using Theorem 2 and Lemma
3.5 in [14] we obtain A∩B ⊆ Int(Cl(ΨG (A)))∩Int(Cl(ΨG (A))) = Int(Cl(ΨG (A)∩
ΨG (B))) = Int(Cl(ΨG (A ∩ B))). Therefore, A ∩ B ⊆ Int(Cl(ΨG (A ∩ B))) and
A ∩ B ∈ τ A.
Proposition 3. [1] Let (X, τ, G) be a grill topological space. Then ΨG (A) 6= φ if
and only if A contains a nonempty τG -interior.
e G (X, τ ) if and
Corollary 4. Let (X, τ, G) be a grill topological space. Then {x} ∈ Ψ
only {x} ∈ τ A
e G (X, τ ), therefore {x} is open in (X, τG ) by Proposition 3. Since
Proof. Let {x} ∈ Ψ
{x} ⊆ ΨG ({x}) and ΨG ({x}) is open in (X, τ ), therefore {x} ⊆ Int(Cl(ΨG ({x}).
Hence {x} ∈ τ A .
Conversely suppose that {x} ∈ τ A . Then {x} ⊆ Int(Cl(ΨG ({x}))) and {x} ⊆
e G (X, τ ).
Cl(ΨG ({x})). Hence {x} ∈ Ψ
Theorem 8. Let (X, τ, G) be a grill topological space. Then τ A consists of exactly
e G (X, τ ) for all B ∈ Ψ
e G (X, τ ).
those sets A for which A ∩ B ∈ Ψ
e G (X, τ ). Now we show that A ∩ B ∈ Ψ
e G (X, τ ). If
Proof. Let A ∈ τ A and B ∈ Ψ
A ∩ B = φ, we are done. Let A ∩ B 6= φ and x ∈ A ∩ B, then x ∈ Int(Cl(ΨG (A))),
since A ∈ τ A . Consider any neighbourhood Ux of x then Ux ∩ Int(Cl(ΨG (A))) is
a neighbourhood of x. Since x ∈ B ⊆ Cl(ΨG (B)), then Ux ∩ Int(Cl(ΨG (A))) ∩
ΨG (B) 6= φ. Let V = Ux ∩ Int(Cl(ΨG (A))) ∩ ΨG (B), then V ⊆ Cl(ΨG (A)). This
implies that Ux ∩ ΨG (A) ∩ ΨG (B) = V ∩ ΨG (A) 6= φ, since ΨG (B) is open. Therefore
x ∈ Cl[ΨG (A) ∩ ΨG (B)] = Cl[ΨG (A ∩ B)]. Hence A ∩ B ⊆ Cl[ΨG (A ∩ B)], therefore
194
Ahmad Al-Omari and Takashi Noiri
e G (X, τ ).
A∩B ∈Ψ
e G (X, τ ) for each B ∈
Next we consider a subset A of X such that A ∩ B ∈ Ψ
e G (X, τ ). We show that A ∈ τ A . Suppose A * Int(Cl(ΨG (A))) then there exists
Ψ
x ∈ A but x ∈
/ Int(Cl(ΨG (A))). Therefore, x ∈ A ∩ [X − Int(Cl(ΨG (A)))] =
A ∩ Cl[X − Cl(ΨG (A))] = A ∩ Cl(H), where H = X − Cl(ΨG (A)). It is obvious that
H is a nonempty open set. Since x ∈ Cl(H), then for all open set Vx containing
x, Vx ∩ H 6= φ. Therefore Vx ∩ ΨG (H) 6= φ, since H ⊆ ΨG (H). This implies
that x ∈ Cl(ΨG (H)) ⊆ Cl(ΨG [H ∪ {x}]) and H ⊆ Cl(ΨG (H)) ⊆ Cl(ΨG [H ∪ {x}])
e G (X, τ ). Now
and hence {x} ∪ H ⊆ Cl(ΨG [H ∪ {x}]). Therefore {x} ∪ H ∈ Ψ
e G (X, τ ), since [{x} ∪ H] ∈ Ψ
e G (X, τ ). We show
by hypothesis A ∩ [{x} ∪ H] ∈ Ψ
that A ∩ [{x} ∪ H] = {x}. Suppose that there exists y ∈ X and x 6= y such that
e G (X, τ ), hence
y ∈ A∩[{x}∪H], then y ∈ A and y ∈ H. Now A = A∩X and X ∈ Ψ
e G (X, τ ). Since y ∈ A ⊆ Cl(ΨG (A), this is contrary to the fact
by hypothesis A ∈ Ψ
e G (X, τ ),
that y ∈ H = X − Cl(ΨG (A)). Thus A ∩ [{x} ∪ H] = {x}. Since {x} ∈ Ψ
then {x} ∈ τ A (by Corollary 4). Hence, {x} ⊆ Int(Cl(ΨG ({x}) = Int(Cl(ΨG (A ∩
[{x} ∪ H]) ⊆ Int(Cl(ΨG (A))). But x ∈ Int(Cl(ΨG (A))) which is contrary to the
fact that x ∈
/ Int(Cl(ΨG (A))). Therefore we have A ⊆ Int(Cl(ΨG (A))) and hence
A ∈ τ A.
Now we refer to the following well known theorem.
Theorem 9. [12] Let (X, τ ) be a topological space. Then τ α consists of exactly
those sets A for which A ∩ B ∈ SO(X, τ ) for all B ∈ SO(X, τ ).
Now from Theorems 6, 8 and 9 we have the following theorem.
Theorem 10. Let (X, τ, G) be a grill topological space, where G is an anti-codense
grill. Then τ A = (τG )α .
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Ahmad Al-Omari and Takashi Noiri
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Ahmad Al-Omari:
Department of Mathematics,
Faculty of Science, Mu’tah University,
P.O.Box 7, Karak 61710, Jordan
E-mail: [email protected]
Takashi Noiri:
2949-1 Shiokita-cho, Hinagu,
Yatsushiro-shi, Kumamoto-ken, 869-5142 Japan
E-mail: [email protected]