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International Journal of Computer Application
Issue 3, Volume 5 (September - October - 2013)
Available online on http://www.rspublication.com/ijca/ijca_index.htm
ISSN: 2250-1797
A Grill view of nearly compact spaces
Karthika.A and I.Arockiarani
Nirmala College for Women
Coimbatore
___________________________________________________________________________
Abstract
In this paper, a new class of topological spaces, namely, the class of G-nearly compact
spaces (G -NC spaces) has been introduced. This class contains the class of G -compact
spaces and is contained in the class of almost G -Compact spaces. Some of the properties of
G -NC spaces have been obtained.
Keywords and phrases: G -near compactness, almost G -compactness, G -cover.
2000 AMS Subject Classification: 54C20, 54C99.
___________________________________________________________________________
1. Introduction
Choquet [4] introduced an attractive theory of Grills, a collection satisfies some
condition in Topological space. The topology equipped with the grill collection is called Grill
topology. This topic has an excellent potential for application in other branches of
mathematics like compactifications, Proximity spaces, different types of extension problems
etc. This subject was continued to study by general topologists Roy and Mukherjee [8], [9] in
recent years. It was probably the two articles of M.K. Singal and Asha Mathur [12], [13] that
initiated the study of nearly Compact spaces in general topology.
The purpose of this paper is to define G-nearly compact spaces and investigete their
basic properties.
Preliminaries
Definition 1.1 [8]
A colletion G of nonempty subsets of a set X is called a grill if
1. A ∈ G and A ⊆ B ⊆ X implies that B ⊆ G, and
2. A ∪ B ∈ G (A, B ⊆ X) implies that A ∈ G or B∈ G.
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Definition 1.2 [9]
Let G be a grill on a topological space (X, τ). A cover {Uα : α ∈ Λ} of X is said to be
a G-cover if there exists a finite subset Λ0 of Λ such that X \ ∪α∈Λ0 Uα ∉ G.
Definition 1.3 [8]
A grill G on a set X is said to be a σ-grill if for any countable collection {𝐴𝑖 : i ∈ I }
of subsets of X,
∪∞
𝑖=1 𝐴𝑖 ∉ G whenever 𝐴𝑖 ∉ G for each i ∈ I .
Definition 1.4 [12]
A subset of a space is said to be regularly open, it is the interior of some closed set or
equivalently, if it is the interior of its own closure. A set is said to be regularly closed if it is
the closure of some open set or equivalently, if it is the closure of its own interior.
Definition 1.5 [1]
A topological space with the grill G is said to be almost G-Compact if for every open
cover V = {Uα : α ∈ Λ} there exists a finite subcollection {Ui : i ∈ I } such that
X \ ∪i=1 n clUi ∉ G.
Definition 1.6 [13]
A space X is said to be semi-regular if every point of the space has a regular open
neighbourhood.
Definition 1.7[13]
A space X is said to be almost-regular if for each point x ∈ X and each neighbourhood
U of x, there exists a neighbourhood V of x such that V ⊂ cl V ⊂ intclU .
Definition 1.8 [12]
A mapping is said to be almost-continuous if the inverse image of every regular open
set is open.
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Definition 1.9[13]
A mapping is said to be almost-open if it sends each regularly open set in to a open
set.
Definition 1.10 [13]
A mapping f:X → Y is said to be strongly continuous if f [clA] ⊂ f [A] for every
subset of A of X.
Definition 1.11 [13]
A mapping f:X → Y is said to be quasi-Compact if the image under f of every open
inverse subset of X is open in Y.
Definition 1.12[12]
A set P is said to be δ-closed if for each point x ∈ P , there exists an open set G
containing x such that (intcl G) ∩ P = υ, or equivalently, for each point x ∈ P , there exists a
regularly open set containing x which has intersection with P. A set is δ-open if and only iff
its compliment is δ-closed.
Definition 1.13 [12]
A point x is a δ-adherent point of a filter-base(filter) F if x ∩ {[F ]δ : F ∈ F }. A
filter-base (filter) F δ-converges to a point x in (X, τ ) if the interior of every closed
neighbourhood of the point x contains an F ∈ F or equivalently, every δ-open set containing
x contains an F ∈ F .
2. Near Compactness in grill
Definition 2.1
Let G be a grill on the topological space (X, τ). Then X is said to be Nearly compact
space if for every open cover V = {Uα : α ∈ Λ} of X, there exists a finite subcollection such
that X \ ∪i=1n intclUi ∉ G.
Note 2.2
The class of G- nearly compact spaces contains the class of G-compact spaces and is
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contained in the class of almost G-Compact spaces.
Remark 2.3
Every G-Compact space is G-Nearly Compact, but the converse is not true and this is
shown in the following example.
Example 2.4
Let τ denote the co-countable topology on an uncountable set X and G be a grill of all
uncountable sets of X. But (X, τ, G) in not G-Compact but G-Nearly Compact, and hence
almost G-Compact.
Definition 2.5
A collection η = {Aα : α ∈ Λ}of sets has the finite intersection property with respect
to grill or simply G.f.i.p provided that the intersection of any finite sub collection belongs
to grill. That is ∩i=1n Ai ∈ G.
Theorem 2.6
In a grill topological space (X, τ, G), with a σ grill G, the following are equivalent:
(a) (X, τ, G) is a G-NC space.
(b) Every basic open cover of X admits a finite subfamily, such that the difference from X to
the union of interiors of closures of whose members does not belongs to the grill G.
(c) Every cover of X by regularly open sets is a G-cover.
(d) Every family of regularly closed sets having the grill finite intersection property (G.f.i.p.)
has arbitrary intersection belongs to the grill.
(e) Every family ν of closed sets having the property that for any finite subfamily
n {Fi : i = 1, 2, 3, ....n} of V , such that ∩i=1n cl intFi ∈ G has intersection belongs to grill.
That is ∩i=1 n Fα ∈ G implies that ∩α ∈ Λ Fα ∈ G.
Proof
(a)⇒ (b): Obvious.
(b)⇒(c): Let U ={Uα : α ∈ Λ} be any regularly open cover of X and let B be a base for τ .
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For each α ∈ I , Uα = ∪α∈Λα {Vλ : Vα ∈ B}. Then V = {Vλ : λ ∈ I} is a basic n open cover of
X. By (b), V has a finite sub-family, such that X \ ∪i=1n intclVi ∉ G. For each Vi ∈ V , there
exists an i ∈ I , such that Vi ⊆ Uαi ⇒ intclVi ⊆ intclUαi = Uαi.
Since G is a σ-grill and also X \ ∪i=1n Uαi ⊆ X \ ∪i=1n intclVi, The finite subfamily
{Uαi : i ∈ I } satisfies the property , X \ ∪i=1n Uαi ∉ G Hence the result (c).
(c)⇒(d): Let F ={Fα : α ∈ Λ} be a family of regularly closed sets having the grill finite
intersection property (G.f.i.p.). If possible, let ∩α∈ΛFα ∉ G. Then {X \ Fα : α ∈ Λ} is a
regularly open cover of X and has a finite sub-cover {X \ Fαi : i = 1, 2, 3...n}, such that
X \ ∪i=1n (X \ Fαi) ∉ G , ⇒ ∩i=1n Fαi ∉ G which is a contradiction to our G.f.i.p. Hence the
result.
(d)⇒(e): Let F be a family of closed sets having the given property. Then {clint F : F ∈ F }
is a family of regular closed sets having the G.f.i.p. By (d), ∩{clint F : F ∈ F } ∈ G. Also
F being closed, clint F ⊂ F and G is a σ grill and Hence ∩{F : F ∈ F } ∈ G.
(e)⇒(a): Let U = {Uα : α ∈ Λ } be any open cover of X. If possible, let there be no finite
subfamily U ={Uαi : i = 1, 2, 3......n} of U such that X \ ∪i=1n intclUαi ∉ G , so that ∩ (X
\int cl(Uαi) ∈ G for any finite subset. ⇒ ∩i=1n cl int (X \ Uαi ) ∈ G. by (e)
∩α∈Λ cl int( X \ Uα) ∈ G. That is, X \ ∪α∈Λ intcl(Uα) ∈ G, which is a contradiction to the fact
that { Uα : α ∈ Λ }is a cover. Hence X is G -Nearly Compact.
Theorem 2.7
Let (X, τ, G) be a semi-regular space with the σ-grill G, then it is G -NC if and only
if it is G -Compact.
Proof
Let (X, τ, G) be a semi-regular space with the σ-grill G, and let U ={Uα : α ∈ Λ} be
any open cover of X. For each x ∈ X , there exists an α x ∈ Λ such that x ∈ Uαx . Since X is
semi-regular, there exists a regularly open set Gx such that x ∈ Gx ⊂ Uαx . Then {Gx : x ∈
X } is a regularly open cover of X and has therefore a finite subcover {G xi : i = 1, 2, 3...n}
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International Journal of Computer Application
Issue 3, Volume 5 (September - October - 2013)
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ISSN: 2250-1797
such that X \ ∪i=1n Gxi ∉ G. Since the grill G is a σ-grill and Gx ⊂ Uαx . Then {Uxi : i = 1,
2, 3......n} is a finite subcollection satisfies X \ ∪i=1n Uxi ∉ G. Hence X is G -Compact. The
converse part is obvious, because every G –Compact space is G -nearly compact.
Theorem 2.8
Let (X, τ, G) be an almost -regular space with the σ-grill G. Then it is an almost G compact space if and only if it is G -nearly Compact.
Proof
Let U ={Uα : α ∈ Λ} be a regularly open cover of an almost regular, almost G Compact space (X, τ, G). For each x ∈ X , there exists an Uαx ∈ Λ such that x ∈ Uαx . By
almost-regularity, there exist exists an open set Vx such that x ∈ Vx ⊂ cl(Vx) ⊂ intcl(Uαx ) =
Uαx .
V ={Vx : x ∈ X } is an open cover of X which is an almost G -Compact space.
Therefore there exists a finite sub-family such that X \ ∪i=1n cl(Vxi ) ∉ G. Since G is a σ-grill
and Vx ⊂ Uαx .Implies that X \∪i=1n Vαxi ∉ G. The converse is Obvious.
Lemma 2.9
Every open cover of regularly closed subset of a G -NC space admits of a finite
subfamily, the difference from X to the interiors of the closures of whose members’ does not
belongs to the grill G.
Proof
Let (X, τ, G) be a grill topological space. U = {Uα : α ∈ Λ} be any open cover of
regularly closed subset Y of an G -NC space X. Then V ∪ (X \Y) is an open cover of X.
There exists a finite subfamily V = {Vλ : λ ∈ I } such that
X \ ∪λ∈I intcl{Vλ ∪ (X \Y )} ∉ G.
Y being regular closed set, intcl (X \Y ) = X \ Y .
If Vλ ⊆ X \Y for no λ ∈ Λ, then V is the required finite subfamily of U , such that
Y \ ∪λ∈Λ Vλ ∉ G. Hence the proof.
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3 Continuous mappings in nearly compact Grill topological spaces
This section examine the notion of continuous mappings in G –NC spaces. Further
we introduce the concept of relative grill with respect to the subspace (A, 𝜏𝐴 ) of the space
(X, 𝜏, G) and study its properties
Theorem 3.1
Let f: (X, τ1, G 1) → (Y,τ2, G 2) be a function from X, which is a G1 - nearly compact
space on to Y. If f is a almost homeomorphism (bijective, almost- continuous, almost-open
mapping) with G2 ⊂ f (G1) and f (G1) is a σ- grill. Then Y is a G2-NC space.
Proof
Here f : (X,τ1, G1) → (Y,τ2, G2) is a almost homeomorphism mapping of an G1-NC
space X on to Y. Let U ={Uα : α ∈ Λ} be any regular open cover of Y. Then f being almost
continuous, U* ={f −1(Uα) : α ∈ Λ} is an open cover of the G1-NC space X. Therefore there
exists a finite subfamily, {f −1(Uα ) : i = 1, 2, 3......n} of U* such that
X \ ∪i=1n intcl(f− 1 (Uαx )) ∉ G1. Now, applying f we get,
f (X )\ f[∪i=1n intcl(f−1 (Uαx ))] ∉ f(G1) since f is on to ,one to one and G 2 ⊂ f(G1)
we have Y \ f [∪i=1n intcl( (Uαx ))] ∉ G 2
That is Y \∪i=1n f [int cl(f−1 (Uαx ))] ∉ G 2 Since f is almost open,
Y \∪i=1n int f (f−1 [cl(Uαx )] ∉ G 2 Since f is almost-continuous,
Y \ ∪i=1n intcl(Uαxi ))] ∉ G 2
Since f is being on to, Y \ ∪i=1n (Uαxi ) ∉ G 2 , Uα’s being regular open.
Thus U has a G 2-cover. Hence Y is G 2-NC.
Theorem 3.2
If f: (X, τ1, G 1)→ (Y,τ2, G2) is a bijective, almost-continuous mapping with G
2
⊂
f(G 1), where X is G 1-Compact and f(G 1) is a σ-grill, then Y is G 2-NC.
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Proof
Let f : (X, τ1, G 1)→ (Y,τ2, G 2) be an bijective, almost-continuous mapping of a G 1Compact space X on to a space Y. Let U ={Uα : α ∈ Λ} be a regular open cover of Y. Then
U ={f −1(Uα); α ∈ Λ} is an open cover of X. Since X is a G 1-Compact space, there exists a
finite sub-family {f −1(Uα ); i = 1, 2, 3.....n} such that, X \ ∪i=1n f((Uαi )) ∉ G 1
Then, f (X \∪i=1n ((Uαi ))) ∉ f(G 1). Since G 2 ⊂ f (G 1) and f is bijective, Y \ ∪i=1n
(Uαi ) ∉ G2. Hence the result.
Lemma 3.3
A space (X, τ, G) is almost G -Compact if and only if every regular open cover,
U ={Uα :α ∈ Λ} has a finite subfamily U* such that X \∪i=1n (cl (Uαi )) ∉ G , where G is a
σ-grill.
Proof
Let (X, τ, G) be a grill topological space. If a space X is almost G -Compact and every
regular open cover is also an open cover, then the condition of the lemma is obvious.
To
prove the converse, let U ={Uα : α ∈ Λ} be any open cover of X. V ={intclUα : α ∈ Λ} is
then regular open cover, so that there exists a finite sub- family of U* such that, there exists a
corresponding sub-family of U* such that X \∪i=1n cl[intcl((Uαi ))] ∉ G.
Then X \ ∪i=1n cl((Uαi )) ∉ G, since G is a σ- grill. Hence X is almost G -Compact.
Theorem 4.2.4
Let f: (X, τ1, G1) → (Y, τ2, G2) be an bijective, almost-continuous mapping of an
almost G1-compact space with G2 ⊂ f(G1) where f(G1 ) is a σ-grill. Then image of almostcontinuous mapping is almost G2-compact.
Proof
Let f: (X, τ1, G1)→ (Y,τ2, G2) be an bijective, almost-continuous mapping of an
almost G1-compact space X on to Y and let {Uα : α ∈ Λ} be any regular open cover of Y.
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{f −1(Uα) : α ∈ λ} is then an open cover of X. Since X is almost G1-Compact, therefore it has a
finite sub-family {f −1(Uα ) : i = 1, 2, 3, ....n} such that X \ ∪i=1n (cl f−1 (Uαi ))] ∉ G1 ,f being
almost continuous cl[f −1(Uα)] ⊂ f −1cl(Uα). Also G2 ⊂ f(G 1), G1 is a σ-grill and f is bijective,
Hence Y \ ∪i=1n (cl(Uαi )) ∉ G2 .Thus Y is almost G2-Compact.
Corollary 3.5
Let f : (X,τ1, G1)→ (Y, τ2, G 2) be an bijective, almost-continuous mapping with G2 ⊂
f(G 1) where G1 is a σ-grill. Then,
1. If X is G1-Compact then Y is almost G2-Compact.
2. If X is G1-nearly compact then Y is almost G2-Compact.
Theorem 3.6
Let f : (X, τ1, G1)→ (Y,τ2, G 2) be an bijective, strongly continuous mapping with G2 ⊂ f(G1)
where f(G1 ) is a σ-grill. Then the image of an almost G1-Compact space is G 2-Compact.
Proof
Let f: (X, τ1, G1) → (Y,τ2, G2) be an bijective, strongly continuous mapping with G2
⊂ f(G1) where f(G 1 )is a σ-grill. Let U ={Uα : α ∈ Λ} be any open cover of Y.
Then U ={f −1(Uα) : α ∈ λ} is an open cover of X. Since X is almost G1-Compact there
exists a finite subfamily {f −1(Uα ) : i = 1, 2, 3, ....n} of U such that
X \ ∪i=1n cl(f−1 (Uαi )) ∉ G1
That is as f is strongly continuous and f(G 1 ) is a σ-grill,
X \ ∪i=1n (f-1(Uαi ) ∉ G1 . Then Y \ ∪i=1n (Uαi ) ∉ G2 . Hence {Uαi : i = 1, 2, 3...n} is a G 2cover and so Y is G2-Compact.
Corollary 3.7
The property of being a G -NC space is preserved under bijective, strongly continuous
mappings with the necessary conditions on the grill.
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Theorem 3.8
An open subset A of a space (X, τ, G) is G -nearly compact if and only if every open
cover of the set has a finite subfamily the interiors of the closures of whose members does not
belong to the grill G where G is a σ-grill.
Proof
Let A be an open G -NC subset of X and let {Uα : α ∈ Λ} be any open cover of X.
Then U ={A ∩ Uα : α ∈ Λ} is then a relatively open cover of A. Let int
A
and cl
A
denote
respectively the interior and closure operator in A. Since A is G -NC, there exists a finite subfamily {A ∩ Uαi : i = 1, 2, 3.....n} of U , such that
A\ ∪i=1n intA clA(A ∩ Uαi ) ∉ G,
then A\ ∪i=1n [A ∩ int cl(A ∩ Uαi )] ∉ G.
As A is open ∪i=1n [A ∩ intcl(A ∩ Uαi )] ⊂ ∪i=1n intcl(Uαi ).
Since G is σ-grill, A\ ∪i=1n intcl(Uαi ) ∉ G.
Conversely, any relatively open cover {Vβ : β ∈ Λ} of A is open in the space also, as
A is open and hence has a finite subfamily {Vβi
: i = 1, 2, 3.......n} such that
A\ ∪i=1n intcl(Vβi ) ∉ G.
That is A\ ∪i=1n [A∩intcl(Xβi )] ∉ G .
Implies that A\ ∪i=1n [A∩intcl(Vβi )] ∉ G.
Then A\ ∪i=1n intA clA(Vβi ) ∉ G.
Hence the result.
Theorem 3.9
Every clopen subset of a G -nearly compact space (X, τ, G) is G -nearly compact
where G is a σ-grill.
Proof
Let Y be any clopen subset of a G -NC space (X, τ, G). Let U be any open cover of Y
so that U ∪ (X \Y ) is an open cover of the G -NC space X. There exists a finite subfamily U
such that X \ ∪i=1n intcl[(Uαi ) ∪ (X \Y )] ∉ G. But int cl(X \Y )= (X \Y ),
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Then X \ ∪i=1n intcl[(Uαi )] ∪ (X \Y ) ∉ G . That is Y \ ∪i=1n intcl[(Uαi )] ∉ G and hence Y is G -NC.
Theorem 3.10
The space (X, τ) with the grill G is G -nearly compact if and only if (X, τ ∗) is G compact.
Proof
Let (X, τ, G) be a G -nearly compact grill topological space. Let C be a basic τ ∗-open
covering of X. Then C is also a τ -covering and hence there exists a finite subfamily,
say {Ci : i = 1, 2, 3....n} such that, X \ ∪i=1n (intcl Ci) ∉ G . Since intclCi = Ci, C has a finite
subcovering and hence (X, τ∗ ) is G -Compact. Conversely, let C be a τ -covering.
Then C ⊆ intclC for all c ∈ C, and therefore {intclC : c ∈ C} is a τ ∗-covering and hence it
has a finite subcollection (here (X, τ ∗) is G -Compact) such that X \ ∪i=1n (intcl Ci) ∉ G. So,
(X, τ) is G -nearly Compact.
Corollary 3.11
For a grill topological space (X, τ, G), the following are equivalent:
(a) X is G -nearly compact.
(b) Every δ-open cover is a G -cover.
(c) Every family of δ-closed sets having finite intersection property with respect to the grill G
(G.f.i.p) has the intersection belongs to the grill G.
(d) Every filter (filter-base) has a δ-adherent point.
Theorem 3.12
A space is G -nearly compact if and only if it is totally co- G -Compact.
Proof
Let (X, τ) be a G -nearly compact grill topological space. Let τB be any co- topology
with the same grill collection G of τ . Let C be a τB -open cover of X.
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Since B ⊆ τ , each X \cl B is a regular open subset of X. Therefore C is a δ-open
cover of X and thus there exists a finite sub-collection such that , X \ ∪i=1n (intclCi) ∉ G.⇒
Since all Ci are regular open X \ ∪i=1n (Ci) ∉ G. ⇒ since B is arbitrary, the co-space is a
G -Compact space for every B.
Conversely, let (X, τ, G) be a totally co- G -Compact space. Consider the co-topology
τ of τ. Then the space (X, τ) is G -Compact. Let C be a regular-open cover of (X, τ). Then for
each C ∈ C, C = intτ clτ C = X \ (X \ intτ clτ C ) = X \clτ (X \ clτ C ). Since X \clτ C ∈ τ, C ∈ τ
and hence C is a τ -open cover of X and thus it is a G -cover ⇒ (X, τ, G) is G -nearly
Compact. Hence the result.
Theorem 3.13
A grill topological space is G2-nearly Compact if and only if it is bijective, almost
continuous image of a G 1-Compact space, with G2 ⊂ f(G 1) and f(G 1) is a σ-grill.
Proof
The bijective, almost-continuous image of a G 1-Compact space is G 2-NC with all the
necessary conditions for grill (with G2 ⊂ f(G 1) and f(G 1) is a σ-grill).
Conversely, let (X, τ, G) be a G -nearly compact space. Now (X, τ ∗, G) is
G -Compact in the view of Theorem 5.4. Consider the identity mapping
i: (X, τ, G) → (X, τ ∗, G). The mapping i is clearly almost continuous as every
τ -regular open set is τ
∗
-open. Thus, there exists a G - Compact space and an almost
continuous mapping such that (X, τ, G) is the bijective, almost continuous image of that G Compact space under the condition G2 ⊂ f(G1) . Hence the result.
Definition 3.15
Let (X, τ, G) be a grill topological space. Let A be a subset of X. Then GA is called
the relative grill of A with respective to the grill G and G A= {A ∩ G : G ∈ G, A ∩ G ≠ }.
Remark 3.16
(1)In general two topologies τ ∗\A and (τ
\A)∗
are not equivalent for any subset A of a grill
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ISSN: 2250-1797
topological space (X, τ, G).
(2) A subset A is called α- G A-nearly compact if (A, τ ∗\A, G A,) is G A-Compact.
(3) A subset A is called G A -nearly compact if (A, (τ \A)∗, G A) is GA-Compact.
(4) In general G A ⊈ G . So, G A-compact and G -compact are independent.
Theorem 3.17
An open subspace of a grill topological space is nearly compact if and only if it is
nearly compact with respect to the relative grill of the subspace.
Proof
Let (A, τ \A, G A) be a subspace of a space (X, τ, G), which is either open or dense in
(X, τ,G). Then for any subset B of A , intAclAB = (int clB) ∩ A. ⇒ τ∗\A = (τ \A) ∗ and hence
the result.
Theorem 3.18
For regular closed subsets, GA-α-near compactness implies G A- near compactness.
Proof
Let A be a G A-α-nearly compact regular closed set of the grill topological space
(X, τ, G). Let τ be the family of (τ
\A)
∗-closed
subsets of A with the finite intersection
property with respect to the grill (G.f.i.p). Since a (τ /A)-regular closed subset of a τ -regular
closed set A is τ -regular closed, τ is a family of τ
∗
\A-closed
subsets of A with G.f.i.p. Since
A is a α- G-nearly compact, (A, τ ∗\A, G A) is G - compact and thus the arbitrary intersection
belongs to the grill G A. Hence the result.
Theorem 3.19
If every subspace of a space (X, τ, G) is almost compact with respect to the relative
grill, then X is semi-irreducible.
Proof
Let every open subspace is almost compact with respect to the relative grill and X is
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ISSN: 2250-1797
not semi-irreducible. Then there is an infinite family {Gα : α ∈ Λ,, Gα ∈ G} of disjoint open
sets.
Let G = ∪{Gα : α ∈ Λ}.Then G is almost G G-Compact subset, for the open cover
{ Gα : α ∈ Λ} has no finite sub-family, such that X \ ∪α∈Λ0 clGα ∉ G G. This is a contradiction
for the fact that G is almost G G-compact. Therefore our claim is true. X is semi-irreducible.
Theorem 3.20
Let (X, τ, G) be a G -nearly compact space. Let A be a closed set such that the
boundary ∂A of A is α- G -nearly compact, then A is also α- G -nearly compact.
Proof
Let {Cα : α ∈ Λ} be a τ ∗\A-open covering of A. For each α ∈ Λ, there exists a
Gα ∈ τ ∗ such that Cα =A ∩ Gα. For each x ∈ A, there exists a Cαx such that x ∈ Cαx and
hence a Gαx such that x ∈ Gαx . Since Gαx ∈ τ ∗, there exists a τ -regular open set Vαx, such
that x ∈ Vαx ⊆ Gαx . Now {Vαx : x ∈ A} ∪{X \A} forms a τ -open cover of the G -nearly
compact space X and hence there exists a finite subfamily {Vαx
: i = 1, 2, 3....n} of
{Vαx : α ∈ Λ} such that,
X \ ∪i=1n (int clVαxi ) ∪ (int cl(X \A)) ∉ G.
Since intcl (X \A)= X \clint (A) and intA ⊆ cl intA,
therefore int A ⊆ ∪{intclVαxi : i = 1, 2, 3....n} = ∪{Vαxi : i = 1, 2, 3....n} ⊆ {Gαxi : i = 1, 2,
3....n}.
Thus int A ⊆ ∪{Gαxi ∩ A : i = 1, 2, 3.....n} = ∪{Cαxi : i = 1, 2, 3, .....n}. Now {Gα ∩ ∂A : α
∈ Λ} is a τ ∗∂A -open covering of the α- G nearly Gαx-compact set ∂A and hence there exists
a finite subfamily { Gαxj : j = 1, 2, 3....m}of {Gα ∩ ∂A : α ∈ Λ} such that ,
∂A\ ∪j=1m {(Gxj ) ∩ ∂A} ∉ G.
Now A= (A\intA) ∪ int A = ∂A ∪ int A,( as A is closed) ⇒
A\ {∪{Cαxj : j = 1, 2, 3, ....m}} ∪{∪{Cαxi : i = 1, 2, 3, ....n}} ∉ G
Hence A is α- G -nearly compact.
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ISSN: 2250-1797
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