Download GENERALIZATION OF COMPACTNESS USING GRILLS A. Karthika

International Journal of Mathematical Archive-4(8), 2013, 284-288
Available online through www.ijma.info ISSN 2229 – 5046
GENERALIZATION OF COMPACTNESS USING GRILLS
A. karthika* & I. Arockiarani
Nirmala college for women, Coimbatore (A.P.), India
(Received on: 11-12-12; Revised & Accepted on: 06-08-13)
ABSTRACT
In this paper we introduce the concept of θ-compactness in terms of the grill G. We also derive Alexender’s subbase
theorem and Tychonoff product theorem for θ-open sets. Further we develop the theory by extending the work to one
point θ-compactification of a θ-T2 and locally compact space.
1. INTRODUCTION
Within the past few decades there has been a significant increase in the applications of topology to fields as diverse as
computers, economics, engineering, chemistry, medicine and cosmology. Topology plays a significant role in space
time geometry as well as in different branches of mathematics. The notions of sets and functions in topological spaces
are highly developed and used extensively in many practical and engineering problems. Topology has wide
applications that range from atomic scale in chemistry to the astronomic scale in Cosmology. As branches of topology,
Digital topology and image processing are some of the direct computer application oriented research areas.
Choquet [1] in 1947 initiated the brilliant notion of a grill which subsequently turned out to be a very convenient tool
for various topological investigations like nets and filters. S. Jafari further introduced and investigated the notion of θcompactness by using θ-open sets introduced by Velicko [10]. According to Choquet [1], a grill G on a topological
space X is a non-null collection of nonempty subsets of X satisfying two 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.
2. PRELIMINARIES
Definition: 2.1[2] A topological space (X, τ) is said to be θ-compact if every cover of X by θ-open sets has a finite sub
cover.
Definition: 2.2 [8] A topological space (X, τ) is said to be a θ-T2 space if any two points are separated by disjoint θ neighbourhoods.
Definition: 2.3 [8] A topological space (X, τ) is said to be quasi θH-closed (QHC, in short) if for every θ open cover U
of X, such that X = ∪{cl U : U∈U 0}. A Hausdorff quasi θH -closed space is called a θH -closed space.
Definition: 2.4[7] Let G be a grill on a topological space (X, τ). A cover {Uα: α ∈Ω} of X is said to be a G-cover if
there exists a finite subset Ω0 of Ω such that X \ ∪α∈Ω0 Uα ∉ G. A cover which is not a G-cover of X will be called Gcover of X
Definition: 2.5 [7] Let G be a grill on a topological space (X, τ). Then (X, τ) is said to be compact with respect to the
grill G or simply G-compact if every open cover of X is a G-cover.
3. G-Θ COMPACTNESS
Definition: 3.1 Let G be a grill on a topological space (X, τ). A θ open cover {Uα: α ∈Ω} of X is said to be a G-θ
cover if there exists a finite subset Ω0 of Ω such that X \ ∪α∈Ω0 Uα ∉ G. A cover which is not a G- θ cover of X will
be called G-θ cover of X.
Corresponding author: A. karthika*
Nirmala college for women, Coimbatore (A.P.), India
International Journal of Mathematical Archive- 4(8), August – 2013
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A. karthika* & I. Arockiarani/ Generalization Of Compactness Using Grills/ IJMA- 4(8), August-2013.
Definition: 3.2 Let G be a grill on a topological space (X, τ). Then (X, τ) is said to be θ compact with respect to the
grill or simply G- θ compact if every θ open cover of X is a G-θ cover.
Remark: 3.3
(1) Every θ compact space (X, τ) is clearly a G-θ compact for any grill G on X.
(2) If we take G = ℘(X) – { φ }, then G-θ compactness of a space (X, τ) reduces to the θ compactness of (X, τ).
(3) If for any grill G on a space (X, τ) is G-θ compact, then (X, τ) is θ compact as τG ⊆ τ and hence is G-θ compact.
(By (1))
Theorem: 3.4 Let G be a grill on a topological space (X, τ). Then (X, τ) is G-θ compact if and only if (X, τG) is G-θ
compact.
Proof: As τG ⊆ τ , it follows that (X, τ) is G-θ compact if (X, τG) is G-θ compact. Conversely, Let (X, τ) be G-θ
compact and {Uα: α ∈Ω} be a basic τ G -θ open cover of X. Then for each α ∈Ω, Uα = Vα \ Aα where Vα ∈ τ -θ
open Aα ∉ G. Then {Vα : α ∈Ω} is a τ- θ open cover of X. Hence by G-θ compactness of (X, τ), there exists a finite
subset Ω0 of Ω such that X \ ∪α ∈Ω0Vα ∉G . Now X \∪α ∈Ω0Uα = X \ ∪α ∈Ω0(Vα \ Aα ) ⊆ X \ (∪α ∈Ω0Vα - ∪α ∈Ω0Aα) ⊆
(X \ ∪α ∈Ω0Vα ) ∪(∪α ∈Ω0 Aα) ∉ G as Aα ∉ G, for all α . Thus (X, τG) is G-θ compact.
Remark: 3.5 For a topological space (X, τ) and a grill G on X, the following implication diagram holds.
(X, τ) is G- compact
(X, τG) is G- compact
(X, τ) is compact
(X, τG ) is compact
(X, τ) is θ compact
(X, τG) is θ compact
(X, τ) is G-θ compact
(X, τG) is G-θ compact
Theorem: 3.6 Let G be a grill on a topological space (X, τ), such that τ–{φ} ⊆ G. If (X, τ) is a G-θ compact then (X, τ)
is Qθ HC.
Proof: Let {Uα: α ∈Ω} be an θ open cover of (X, τ). Then by G-θ compactness of X, there exists a finite subset Ω0 of
Ω such that X \ ∪α ∈Ω Uα ∉ G. Then int[X \ ∪α ∈Ω Uα] = φ. For otherwise int [X \ ∪α ∈Ω Uα] ∈ τ – {φ} and hence X \
∪α ∈Ω Uα ∈G , a contradiction. Hence X = ∪α ∈Ω cl Uα and X is Qθ HC.
Definition: 3.7 Let (X, τ) be a topological space and G be a grill on X. then the space X is said to be G- θ regular if for
any θ- closed set F in X with x ∉F, there exist disjoint θ-open sets U and V such that x ∈U and F\V ∉ G.
Theorem: 3.8 Let G be a grill on a θHausdorff space (X, τ). If (X, τ) is G-θ compact then it is G- θ regular.
Proof: Let F be a closed subset of X and x ∉F. By θ-T2 property of X for each y∈F there exist disjoint θ-open sets Uy
and Vy such that x∈Uy and y∈Vy . Now {Vy: y ∈F}∪{X\F} is a θ-open cover of of X. Then by G-θ compactness X
\ [ (∪ni=1 Vyi )∪(X \ F)] ∉ G. Let G = ∪ni=1cl Vyi and H = ∪ni=1 Vyi. Then G and H are disjoint θ-open sets and hence
(X, τ) is G-θ regular.
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Corollary: 3.9 Let G be a grill on a θ-T2 space such that τ – {φ} ⊆ G. If (X, τ) is G-θ compact then it is θH-closed and
G-θ regular.
Theorem: 3.10 Let (X, τ) be an θ H-closed and G be a grill on X. If (X, τ) is G-θ regular then (X, τ) is G-θ compact.
Proof: For the space X using the Qθ HC and G-θ regularity we can find that every θ-open cover is a G-θ cover.
Corollary: 3.11 Let G be a grill on a T2 space (X, τ) such that τ-{φ} ⊆ G. Then (X, τ) is G-θ compact iff (X, τ) is Qθ
HC and G-θ regular.
• Our aim is to derive the result that the catesian product of G-θ compact spaces is again G-θ compact. For that we
need to find an analogue of Alexander’s θ subbase theorem for G-θ compact.
Theorem: 3.12 Let U be an θ-open subbase for a topological space (X, τ). If X has an θ-open G θ-cover then there is a
G-θ cover of X, which consists of elements of U.
Proof: Let C be the collection of all θ-open G - θ covers of X. Then by hypothesis C is nonempty. Let {P α} be a
linearly ordered subset of C .Then by Zorn’s lemma C contains a maximal element P. Thus if H is θ-open and H ∉ G,
then there exist finitely many G1, G2 ,….. Gn ∈P such that X \ H∪ G1∪G2……∪Gn ∉ G.
Next we show that the family of θ-open sets which does not belong to P form a filter.
For this let H1,H2∈τ and H1,H2 ∉ P .Then X \ (H1∪G1∪G2∪…∪Gn) = A1∉ G and H \ (H2∪V1∪V2∪…∪Vm) =
A2 ∉ G, for some finite sub collections {G1,G2,…Gn} and {V1,V2,…Vm} of P. Consider B=X \ [ (H1∩ H2)
(G1∪G2……∪Gn) ∪(V1∪V2∪…∪Vm)]. Then B ⊆ A1 ∪A2. Since A1 ∪A2 ∉ G, we have B ∉ G .Thus (H1∩H2)
∈P. Next let H ∉ ∉ P and H ⊆ G , where G and H are θ-open sets. Then X \ (H1∪G1∪G2∪…∪Gn) ∉ G for finitely
many G1,G2,…Gn ∈P. Thus X \ (G∪G1∪G2∪…∪Gn) ∉ P and hence G ∈τ \P.
Finally to complete the proof it is sufficient to show that U ∩ P is a G-θ cover. Let x∈X. Since P is an θ-open cover
of X, there exists a G ∈P such that x∈G. Since U is a subbase for (X, τ), there exist H1, H2,…Hn ∈U such that x
∈ H1∩H2∩…∩Hn ⊆ G. Then there exists an Hi (for some i =1,2,….,n) such that Hi ∈P. For otherwise, if Hi ∉ P for
all i=1, 2, 3…., n, then ∪i=1 n Hi ∉ P as the family of all θ-open sets not in P forms a filter. Thus G ∉ P, a
contradiction and so U ∩P is a G-θ cover.
Corollary: 3.13 Let G be a grill on a topological space (X, τ). Then X is G-θ compact iff there exists a subbase S
which is a G-θ cover.
Theorem: 3.14 Let {X α : α ∈Ω}be a family of topological spaces, and Gα be a grill on the Cartesian product space X=
∏ α ∈Ω X α such that G ⊆ π -1 α (G α) for each α ∈Ω, where π α: X → X α is as usual αth projection map. If Xα is Gαcompact for each α ∈Ω, then X is G-θ compact.
Proof: Let U be a subbase θ-open cover of X. It is sufficient to find subset {U1, U2,…Un} of U such that X \ ∪i=1n Ui
∉G . Let α0 ∈Ω and Uα0 denote the family of all those subsets V of Xα0 such that π -1 α (V) ∈U. We claim that for at
least one α ∈Ω, Uα is a covering for Xα . If not, then by choosing a point in X not covered by U, which contradicts the
fact that U is a cover of X. Thus there exist a β0 ∈Ω such that Uβ0 is a cover of Xβ0. Then we can find finitely many
U1β0, U2β0,….Unβ0 ∈Uβ0 such that (Xβ0 \ ∪i=1n Ui β0)∉G . By considering Ui = π -1 β0(Ui β0) ∈U ,we see that (X \
∪i=1n Ui)∉G. Thus X is G-θ compact.
4. ONE-POINT Θ - COMPACTIFICATION VIA GRILLS
The intent of this section is constructing of one-point θ –compactification of a locally compact θ-Hausdorff space.
Theorem: 4.1 Let (X, τ) be a topological space and G be a grill on X. Let p be an object, not in X, and put X* =X
∪{p}. Then the map fθ: ℘(X*) →℘(X*) defined by
clθ A ,
if cl A ∉ G, for A ⊆ X
fθ =
clθ A ∪{p} ,
if cl A ∈G, for A ⊆ X
clθ (A \ {p})∪{p}, if p∈A
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is a kuratowski closure operator, inducing a topology τ* such that
(a) Every τ-θ open set in X is τ*-θ open .
(b) If U ( ⊆ X* ) is τ* θ-open then U ∩ X is τ- θ open.
Proof: By considering three cases A, B ⊆ X, A⊆ X and p ∈B, also p∈A and p∈B, we can prove the two
kuratowski’s closure axioms
(1) fθ (A ∪B) = fθ (A)∪ fθ (B),
(2) fθ ( fθ (A))= fθ (A) .
Theorem: 4.2 Let G be a grill on a θT1 space (X, τ) such that for every x ∈X.{x} ∉ G. Adjoin to X a new object
p ∉ X. Then there exists a topology on X* =X∪{p} satisfying the following properties:
(a) X* is θT1
(b) X is dense in X *.
Proof: Constructing a new extended space (X *, τ*) as in the theorem 4.1, we can prove the theorem. Now for any
x ∈X, f(x) = x as cl ({x}) = {x}∉G , and f(p) = cl({p} \ {p})∪{p}.
This proves (a). Again, since cl X = X∈G, f(X) = cl X∪{p}= X ∪{p} = X *, proving (b).
Theorem: 4.3 Let G be a grill on a θHausdorff space (X, τ) such that for every x∈ X, {x}∉G .If for every point x ∈X
there is an open neighbourhood U of x such that clU∉G, then one can construct a one-point extension X* = X ∪{p}
where p ∉ X, satisfying the following properties:
(i) X* is θHausdorff .
(ii) X is dense in X *.
Proof: We consider again the one-point extension space (X*, τ*) of theorem 4.1. We first note that as cl X∈G, τ* -cl X
= X*. Let us consider x, y two distinct points of X. by θHausdorff ness of (X, τ), x and y are separated by sets U,V
which are θ-open in X and hence are θ-open in X*.By the hypothesis, for any x∈ X there is a τ- θ open neighbourhood
U of x such that clθU ∉ G. Let N= X* \ U. Since clθU ∉ G, we have f(U) = U. Thus N is a θ-open neighbourhood of p
in X*. Consequently, U and N are the disjoint τ* - θ open neighbourhoods of x and p respectively in X*. Hence X* is
θHausdorff.
Corollary: 4.3 Let G be a grill on a θT2 space (X, τ) such that for every point x ∈X. {x}∉ G. If for every point x ∈X
there is an θ-open neighbourhood U of x such that cl U∉ G, then one can construct a one-point extension X *= X ∪{p}
then X* is θT2.
Theorem: 4.4 Let (X, τ) be a non- θ-compact, locally compact, θ-T2 space. By adjoining a new point p ∉X to X, one
can construct an extension space X *= X ∪{p} having the following properties:
(1) X* is θ-Hausdorff.
(2) X is dense in X *.
(3) X* is θ-compact.
Proof: Let G .be the family of subsets of X whose closures in X are not θ-compact in X. It is easy to verify that G is a
grill on X such that for each x∈X, {x} ∉ G. For each x ∈ X, by local compactness of X, there is a θ-open
neighbourhood of U of x such that clθU is θ compact. That is clθU ∉ G. Then G satisfies the conditions of the previous
theorem and so (1) and (2) can easily proved.
To prove (3) Let U = {Uα: α ∈Ω} be any cover of X * by θ-open sets of X*. Then for some α 0∈Ω, p∈ Uα0. Then
f (X *\ Uα0) = ( X *\ Uα0) and p ∉Uα0 implies that clθ (X *\ Uα0) ∉G .That is (X *\ Uα0) ∉G .Therefore clθ ( X *\ Uα0) is
is θ compact in X. Since {Uα ∩ X : α ∈Ω} is an θ-open cover of of X. clθ( X *\ Uα0) ⊆∪ni=1(Uαi) ∩ X, for finitely
many sets Uα1 ,Uα2, Uα3, ….Uαn of U. Then X* = (∪ni=1(Uαi) )∪Uα0 , and hence X* is θ compact.
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Source of support: Nil, Conflict of interest: None Declared
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