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Int Jr. of Mathematics Sciences & Applications
Vol. 2, No. 2, May 2012
Copyright  Mind Reader Publications
ISSN No: 2230-9888
www.journalshub.com
A VIEW ON NETS IN TOPOLOGICAL SPACES
VIA b-OPEN SETS
G. Vasuki , E. Roja and M.K. Uma
Department of Mathematics,
Sri Sarada College for Women,
Salem - 636 016
Tamil Nadu.
Abstract
In this paper we define nets, and discuss its convergency in topological space. We prove that
the nets are adequate to describe the closure of a set and the continuity of a function in
topological space via b-open. The properties of b-compact spaces by using nets, filterbases, bcomplete accumulation points are studied.
Keywords : Nets, b-interior, b-closure, b-compact space, b-complete accumulation point.
2010 AMS subject classification primary : 54A05, 54A10, 54A20
1. INTRODUCTION
Topology is Analysis in a general setting. The concept of b-open sets was studied by Andrijevic [1].
Using the concept of b-open sets, the nets and their convergency, closure and continuity are studied. Some
characterizations
of
b-compact spaces in terms of nets and filterbases, b-complete accumulation points are also studied.
2. PRELIMINARIES
Definition 2.1
Let R be a usual metric (i) Let A ⊂ R be a non empty set and x ∈ R. x ∈ cl( A ) iff there is a sequence
in A converging to x. (ii) A function f : R → R is continuous at x0 iff ( xn ) → x0 ⇒ ( f ( xn ) ) → f ( x0 ) for every
sequence ( xn ).
Remark 2.1
The domain of definition D of a net is taken to be a directed set.
Definition 2.2
A set D is a directed set if there is a relation ≤ on D satisfying
(i) ≤ is reflexive (ii) ≤ is transitive and (iii) if λ1, λ2 ∈ D, then there is some λ3 ∈ D with λ1 ≤ λ3 and λ2 ≤ λ3.
Definition 2.3 [8]
A net is a pair ( S, ≤ ) such that S is a function and ≤ directs the domain of S.
Remark 2.2 [8]
A net is sometimes called a directed set.
Definition 2.4
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G. Vasuki, E. Roja and M.K. Uma
Let ( X, τ ) and ( Y, ν ) be any two topological spaces. Then the following are two well known results :
(i) Let A ⊂ X be nonempty and x ∈ X. Then x ∈ cl ( A ) iff there is a net in A convenging to x. (ii) A function f
: X → Y is ( τ, ν ) continuous at x0 iff whenever the net ( xλ ) → ( x0 ) in X, the net ( f ( xλ ) ) → f ( x0 ) in Y.
Remark 2.3
Throughout the section, the directed set is denoted by D and the poset by P.
Remark 2.4 [4]
Let ( X, µ ) be a topological space and Cµ ( A ) is the µ-closure of A. It is clear that, in a topological
space ( X, µ ) x ∈ Cµ ( A ) iff
G I A ≠ φ for every µ-open set G containing x. Every µ-open set containing x ∈
X is called a neighbourhood of x and the family of neighbourhoods of x is denoted by µ ( x ). If µ ( x ) ≠ φ, then
the following statements hold.
(a)
x ∈ U for every U ∈ µ ( x )
U V∈µ(x)
(b)
If U, V ∈ µ ( x ), then U
(c)
If y ∈ U ∈ µ ( x ), then U ∈ µ ( y ).
Definition 2.5 [4]
Let ( X, µ ) and ( Y, λ ) be any two topological spaces. A function
f : ( X, µ ) → ( Y, λ ) is said to be ( µ, λ )-continuous if the inverse image of every
λ-open subset of Y is a µ-open subset of X.
Definition 2.6 [5]
Let ( X, µ ) be a topological space. Then ( X, µ ) is called a T1-space if for every pair of points atleast
one point has a neighbourhood not containing the other.
Definition 2.7 [5]
A topological space ( X, µ ) is called a T2-space if every pair of distinct points have disjoint
neighbourhoods.
Definition 2.8 [1]
A subset A of X is called a b-open set if A ⊆ cl ( int ( A ) ) U int ( cl ( A ) ). The complement of a bopen set is called a b-closed set. The family of b-open sets in a topological space ( X, T ) will be denoted by B 0
( X ).
Definition 2.9 [10]
Let ( X, τ ) and ( Y, σ ) be any two topological spaces. Let the interior and closure of a set A is
denoted by int ( A ) and cl ( A ). A point x ∈ X is called a
θ-adherent point of A if A
Definition 2.10 [10]
The set of
θ cl ( A ).
I cl ( U ) ≠ φ for every open set U of X containing x.
all
θ-adherent points of
A
is
called
the
θ-closure
of
A
denoted
by
Definition 2.11 [10]
The subset A is called θ-closed if A = θcl( A ). The complement of a
θ-closed set is called θ-open. The collection of all θ-open (respectively, θ-closed) sets is denoted by θO( X, τ ) (
resp. clθ( X, τ ) ).
Definition 2.12 [8]
A filter F in a set X is a family of non-void subsets of X such that
566
A VIEW ON NETS IN TOPOLOGICAL SPACES…
(i)
(ii)
The intersection of two member of F always belongs to F.
If A ∈ F and A ⊂ B ⊂ X, then B ∈ F.
Definition 2.13 [9]
Let A be a directed set. A net ξ = { Xα | α ∈ Λ } θ-accumulates at a point
x ∈ X if the net is frequently in every U ∈ θO ( X, x ). The net ξ θ-converges to a point x of X if it is eventually
in every U ∈ θO( X, x ).
Definition 2.14 [9]
A filterbase
x∈
Θ
=
{
Fα
/
α ∈
Γ
}
θ
-
accumulates
at
a
point
x
∈
X
if
I θcl( Fα ). For a given set S with S ⊂ X, a θ-cover of S is a family of θ-open subsets Uα of X for each α
α ∈Γ
∈ I of X such that S ⊂
U Uα.
α ∈I
Definition 2.15 [9]
A filterbase θ = { Fα / α ∈ Γ } θ-converges to a point x in X for each
U ∈ Θ O ( X, x ), there exists an Fα in Θ such that Fα ⊂ U.
Definition 2.16 [2]
A point x in a space X is said to be a θ-complete accumulation point of a subset S of X if card (S
I U)
= card (S) for each U ∈ θ O (X, x) where card (S) denote the cardinality of S.
Definition 2.17 [8]
Zorn’s Lemma :
If each chain in a partially ordered set has an upper bound, then there is a maximal element of the set.
Definition 2.18 [9]
A space X is said to be θ-compact if every θ-open cover of X has a finite subcover which covers X.
3. NETS AND THEIR CONVERGENCY VIA b-OPEN
Definition 3.1 [1]
The largest b-open set contained in A is called the b-interior of A and is denoted by ib ( A ). The
smallest b-closed set containing A is called the b-closure of A and is denoted by Cb ( A ).
Remark 3.1
Let (
x ∈ Cb ( A ) iff
X,
µ
)
be
a
topological
space.
It
is
clear
that
in
a
topological
space
G I A ≠ φ for every b-open set G containing x. Every b-open set containing x ∈ X is called a
b-neighbourhood
of
x
b-neighbourhoods of x is denoted by µb ( x ).
and
the
family
of
all
Remark 3.2
If µb ( x ) ≠ φ, then the following statements hold:
(a)
x ∈ U, for every U ∈ µb ( x )
(b)
If U, V ∈ µb ( x ) then U
U V ∈ µb ( x )
(c)
If y ∈ U ∈ µb ( x ), then U ∈ µb ( y )
Definition 3.2
Let ( X, µ ) and ( Y, λ ) be any two topological spaces. A function
f : ( X, µ ) → ( Y, λ ) is said to be b continuous if the inverse image of every b-open subset of Y is a b open
subset of X.
Definition 3.3
Let ( X, µ ) be a topological space. Then ( X, µ ) is called a b-T1 space if for every pair of points atleast
one point has a b-neighbourhood not containing the other.
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G. Vasuki, E. Roja and M.K. Uma
Definition 3.4
Let ( X, µ ) be a topological space. Then ( X, µ ) is called a b-T2 space if for every pair of distinct
points have disjoint b-neighbourhoods.
Definition 3.5
Let ( X, µ ) be a topological space and ( P, ≥ ) be a poset. A net in X is a function f : P → X. We denote
the image of λ ∈ P under f by fλ and the net will be denoted ( fλ ).
Definition 3.6
Let ( X, µ ) be a topological space. A net ( fλ ) is said to be eventually in
b-neighbourhood U if there exists a λ0 ∈ P such that ( fλ ) ∈ U for every λ ≥ λ0.
Definition 3.7
A net ( fλ ) →
/ x ∈ X if there exists a b-neighbourhood U of x such that ( fλ ) is not eventually in U and
( fλ ) → x, if otherwise. If ( fλ ) → x, x is called a limit of ( fλ ).
Definition 3.8
A net ( fλ ) is said to be frequently in b-neighbourhood U if for every λ ∈ P, there exists a λ1 ∈ P such
that λ1 ≥ λ and (
f λ ) ∈ U.
1
Definition 3.9
x is called a b limit point of ( fλ ) if it is frequently in every b-neighbourhood of x.
Proposition 3.1
Let ( X, µ ) be a topological space. Then b-limit of every constant net is unique iff ( X, µ ) is a b-T1
space.
Proof
Let ( X, µ ) be a b-T1 space and x, y ∈ X. such that x ≠ y. The constant
net x, x, … converges to y implies that x is eventually in every b-neighbourhood of y which implies that x ∈ U
for every U ∈ µb( y ) which is a contradiction to hypothesis. Conversely, x, x, x, … does not converge to y for
any y ≠ x implies that there exists a b-neighbourhood of y not containing x. Hence every point of X has a bneighbourhood not containing the other and so ( X, µ ) is a b-T1 space.
Proposition 3.2
Let ( X, µ ) be a topological space and x, y ∈ X such that x ≠ y and ( fλ ) be a
net in X. Then ( fλ ) → x ⇒ ( fλ ) → y iff x ∈
I { U / U ∈ µb ( y ) }.
Proof
Assume that ( fλ ) → x, ( fλ ) → y. This implies that the, constant net x, x, …x which converges to x
converges
to
y
which
implies
that
x
is
eventually
in
every
b-neighbourhood of y. This implies that x ∈ U for every U ∈ µb ( y ), and hence,
x∈
I { U / U ∈ µb ( y ) }. Conversely suppose that x ∈ I { U / U ∈ µb ( y ) } which implies that every U ∈
µb ( y ) is a b-neighbourhood of x. Now, if ( fλ ) is a net such that ( fλ ) → x, which implies that ( fλ ) is
eventually in every b-neighbourhood of x and so ( fλ ) is eventually in every b-neighbourhood of y. Therefore, (
fλ ) → y.
4. NETS DESCRIBE b-CLOSURE AND b-CONTINUITY
Proposition 4.1
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A VIEW ON NETS IN TOPOLOGICAL SPACES…
Let ( X, µ ) and ( Y, λ ) be any two topological spaces. A function g : X → Y is
b-continuous at x0 ∈ X iff for every net ( fλ ) → x0, the net ( g ( fλ ) ) → g ( x0 ).
Proof
Suppose that g : X → Y is b-continuous at x0 ∈ X. If V is a b-neighbourhood of f( x0 ) there exists a bneighbourhood U of x0 such that g ( U ) ⊂ V. ( fλ ) → x0 implies that ( fλ ) is eventually in U which implies that
( g (fλ ) ) is eventually in g( U ) ⊂ V. Hence ( g ( fλ ) ) → g ( x0 ).
Conversely, suppose that every net ( fλ ) → x0, the net ( g ( fλ ) ) → g ( x0 ). If
g is not b-continuous at x0, then there exists a b-neighbourhood V of g( x0 ) such that
g( U ) ⊆
/ V for any b-neighbourhood U of x0. For each b-neighbourhood U of x0, we can find fU ∈ U and g( fU )
∉ V. Then f : µ( x0 ) → X defined as f( U ) = fU ∈ U is a net in X such that ( fU ) → x0 and ( g ( fU ) ) →
/ g( x0 )
which is a contradiction. Hence g is b-continuous at x0 ∈ X.
5. CHARACTERIZATION OF b-COMPACT SPACES
Definition 5.1
Let ( X, T ) and ( Y, S ) be any two topological spaces. A point x ∈ X is called
b-adherent point of A if A
points
of
A
by bcl ( A ).
I cl ( U ) ≠ φ for every b-open set U of X containing x. The set of all b-adherent
is
called
the
b-closure
of
A
and
denoted
Definition 5.2
A subset A is called b-closed if A = bcl ( A ) The complement of a b-closed set is called b-open. The
collection of all b-open (resp.b-closed) sets denoted by BO( X, T ) (resp. bcl ( X, T ) ).
Definition 5.3
Let ( X, T ) be a topological space. A point x is an b-accumulation point of a subset A iff every bneighbourhood of x contains points of A other than x.
Definition 5.4
Let Λ be a directed set. A net ξ = {Xα / α ∈ Λ } b-accumulates at a point x ∈ X if the net is frequently
in every U ∈ BO ( X, x ). The net ξ is said to be b-converges to a point x of X if it is eventually in every U ∈
BO ( X, x).
Definition 5.5
A filterbase
x∈
B
=
{
Fα
/
α
∈
Γ
}
b-accumulates
at
a
point
x
∈
X
if
I bcl (Fα).
α∈Γ
Definition 5.6
Given a set S with S ⊂ X, a b-open cover of S is a family of b-open subsets Uα of X for each α∈ I of X
such that S ⊂
U Uα.
α∈I
Definition 5.7
A filterbase B = { Fα / α ∈ Γ } b-converges to a point x in X for each
U ∈ BO ( X, x ) there exists an Fα in B such that Fα ⊂ U.
Definition 5.8
A space X is said to be b-compact if every b-open cover of X has a finite sub collection which covers
X.
Definition 5.9
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G. Vasuki, E. Roja and M.K. Uma
A point x in a space X is said to be a b-complete accumulation point of a subset
S of X if card ( S
I U ) = card ( S ) for each U ∈ BO ( X, x), where card (S) denotes the cardinality of S.
Proposition 5.1
A space X is b-compact iff each infinite subset of X has a b-complete accumulation point.
Proof
are
Let the space X be a b-compact and S an infinite subset of X. Let K be the set of points x in X which
not
b-complete
accumulation
point
of
S.
Hence
for
each
point
x in K, we are able to find U(x) ∈ BO (X, x) such that card ( S
I U ( x ) ) ≠ card ( S ). If K is the whole space
X, then B = { U ( x ) / x∈ X } is a b-open cover of X.
By hypothesis X is b-compact, so there exists a finite subcover ψ = { U ( xi ) }, where i = 1, 2, 3 … n
such
that
S
⊂
U
{
card ( S ) = max { card ( ( U ( xi )
complete accumulation point.
(
U
(
xi
)
I
S
)
/
i
=
1,
2,
…
n
}.
Then
I S ) } i = 1, 2, … n } which is a contradiction. Therefore, S has a b-
Conversely, assume that X is not b-compact and that every infinite subset S ⊂ X has a b-complete
accumulation point in X. Then, there exists a b-open cover B with no finite subcover. Let δ = min { card ( Φ ) /
Φ ⊂ B, where φ is a b-open cover of X }. Fix ψ ⊂ B for which card ( ψ ) = δ and
U { U / U ∈ ψ } = X. Let N
denote the set of natural numbers. Then by hypothesis δ ≥ card ( N ).
By well-ordering ψ be some minimal well-ordering “∼”. Suppose that U is any member of ψ. By
minimal well-ordering “∼” we have card ( { V / V ∈ ψ, V ∼ U } )
< card ( { V/ V ∈ ψ } ). Since ψ cannot have any subcover with cardinality lesser than
δ for each U ∈ ψ, we have X ≠
U { V / V ∈ ψ, V ∼ U }. For each U ∈ ψ, choose a point x ( U ) ∈ X − U { V
U { x ( V ) } \ V∈ ψ, V ∼ U }. This always able to do this. If not, one can choose a cover of smaller quantity
from ψ. If H ( x ) = x (U) / U ∈ ψ }, then proof will be complete if we show that H has no b-complete
accumulation point in X. Suppose that z is a point of the space X. Since ψ is a b-open cover of X then z is a
point of some set W in ψ. Since W∼U, x ( U ) ∈ W. Now, T = { U / U ∈ ψ and
x ( U ) ∈ W } ⊂ { V \ V∈ ψ, V ∼ W }. But card ( T ) < δ. Therefore, card (H
I W) < δ. But card ( H ) = δ ≥
card
(
N
)
since
for
two
distinct
points
u
and
w
in
ψ,
we
have,
x ( u ) ≠ x ( w ). This means that H has no b-complete accumulation point in X, which is a contradiction to our
assumption. Therefore X is b-compact.
Proposition 5.2
For a space X the following statements are equivalent:
a)
X is b-compact
b)
Every net in X with an well-ordered directed set Λ as domain, b-accumulates to some point of X.
Proof
(a) ⇒ (b)
Suppose ( X, T ) is b-compact and ξ = { xα/ α ∈ Λ } be a net with an well-ordered directed set Λ as
domain. Assume that ξ has no b-accumulation point in X. Then for each point x in X, there exists V(x) ∈ BO (
X, x ) and an α ( x ) ∈ Λ such that V ( x ) I { xα / α ≥ α ( x ) } = φ which implies that { xα / α ≥ α ( x ) } is a
subset
of
X – V(x). Then the collection C = { V(x) / x∈X } is a b-open cover of X. By the hypothesis of the theorem, X is
b-compact and so C has a finite subfamily {V(xi)}, where i = 1, 2, … n such that X =
570
U {V(xi)}. Suppose that
A VIEW ON NETS IN TOPOLOGICAL SPACES…
the corresponding elements of Λ be { α (xi) }, i = 1, 2, … n. Since Λ is well ordered and { α ( xi ) } is finite, the
largest element of { α (xi) } exists. Let it be { α (xl) }. Then for γ ≥ { α (xl) }, we have
{ xδ / δ ≥ γ } ⊂
n
n
i=1
i=1
I (X−V (xi) ) = X− I V (xi) = φ, which is not possible. This shows that ξ has atleast one b-
accumulation point in X.
(b) ⇒ (a)
It is enough to prove that each infinite subset has a b-complete accumulation point by utilizing
Proposition 5.1. Suppose that S ⊂ X is an infinite subset of X. According to Zorn’s lemma, the infinite set S can
be well-ordered. This means that we can assume S to be as a net with domain which is a well-ordered index set.
Hence, S has a b-accumulation point z. Therefore, z is a b-complete accumulation point of S. Hence X is bcompact.
Proposition 5.3
A space X is b-compact iff each family of b-closed subsets of X with the finite intersection property
has a non-empty intersection.
Proof
The proof is obvious.
Proposition 5.4
A space X is b-compact iff each filterbase in X has atleast one b-accumulation point.
Proof
Fα′s
Suppose that X is b-compact and B = {Fα / α ∈ Γ } be a filterbase in it. Since all finite intersections of
are
non
empty,
it
follows
that
all
finite
intersections
of
bcl (Fα)’s are also non-empty. From Proposition (5.3) that
I bcl (Fα) is non empty, which means that B has
α∈Γ
atleast one b-accumulation point.
Conversely, suppose B is any family of b-closed sets. Let each finite intersection of each family of bclosed sets be non-empty. The sets Fα with their finite intersection establish a filterbase B. Therefore B, baccumulates to some point z in X. Hence, z ∈
I (Fα). Hence by Proposition 5.2 that X is b-compact.
α∈Γ
Proposition 5.5
A space X is b-compact iff each filterbase on X with atmost one b-accumulation point is b-convergent.
Proof
Suppose that X is b-compact, x be a point of X and B be a filter base on X. The
b-adherence of B is a subset { x }. Then the b-adherene of B is equal to { x } by Proposition 5.4. Assume that
there
exists
V
∈
BO
(X,
x)
such
that
for
all
F
∈
B,
FI ( X – V )
is non-empty. Then ψ = { F − V / F ∈ B } is a filterbase on X. Hence the
b-adherence of ψ is non-empty. However
I bcl ( F – V ) ⊂ I bcl ( F ) ) ⊂ ( X – V )
F∈B
F∈B
= { x } I ( X – V ) = φ, which is a contradiction. Hence for each V ∈ BO ( X, x ), there exists F ∈ B with F ⊂
V. This shows that B b-converges to x.
To prove the converse, it is sufficient to show that each filterbase in X has atleast one b-accumulation
point.
Assume
that
B
is
a
filterbase
on
X
with
no
b-accumulation point. By hypothesis B b-converges to some point z in X. Suppose,
Fα is an arbitrary element of B. Then for each V ∈ BO ( X, z ) there exists Fβ ∈ B, such that Fβ ⊂ V. Since B is a
571
G. Vasuki, E. Roja and M.K. Uma
filterbase, there exists a γ such that Fγ ⊂ Fα
I Fβ ⊂ Fα I V, where Fγ non-empty, which means that Fα I V is
non-empty for every V ∈ BO( X, Z ) and correspondingly for each α, z is a point of bcl(Fα). Hence z ∈ I bcl
α
(Fα). Therefore, z is a b-accumulation point of B which is a contradiction. Hence, X is
b-compact.
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