Download g b-Closed Sets in Topological Spaces 1 Introduction

Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 27, 1305 - 1312
g ∗b-Closed Sets in Topological Spaces
D. Vidhya
Department of Science and Humanities
Karpagam College of Engineering
Coimbatore-32, India
vidhya [email protected]
R. Parimelazhagan
Department of Science and Humanities
Karpagam College of Engineering
Coimbatore-32, India
pari [email protected]
Abstract
In this paper, the authors introduce a new class of sets called generalized* b-closed sets in topological spaces (briefly g ∗ b-closed set). Also we
study some of its basic properties and investigate the relations between
the associated topology.
Mathematics Subject Classification: 54A05
Keywords: g ∗ b -closed set, gb-closed set
1
Introduction
N.Levine[4] introduced the notion of generalized closed (briefly g-closed) sets
in topological spaces and showed that compactness, countably compactness,
para compactness and normality etc are all g-closed hereditary. J.Dontchev[3],
H.Maki, R.Devi and K.Balachandran[6], A.S.Mashhour, M.E.Abd-El-Monsef
and SN.El-Deeb[7], D.Andrijevic[1] and N.Nagaveni[8] introduced and investigated the concept of generalized semi-preclosed sets, generalized α-closed
sets, preclosed sets, semi-preclosed sets and weakly generalized closed sets respectively. D.Andrijevic[2], introduced a class of generalized open sets in a
topological space called b-open sets. A.A.Omari and M.S.M.Noorani[10] introduced and studied the concept of generalized b-closed sets(briefly gb-closed) in
topological spaces.
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In this paper,we introduce a new class of sets called g ∗ b -closed set which is
between the class of b-closed sets and the class of gb-closed sets.
Throughout this paper (X, τ ) and (Y, σ)(or simply X and Y) represents the
non-empty topological spaces on which no separation axioms are assumed, unless otherwise mentioned. For a subset A of X ,cl(A) and int(A) represent the
closure of A and interior of A respectively.
2
Preliminaries
Before entering into our work we recall the following definitions which are due
to Levine.
Definition 2.1 [7]:Let A subset A of a topological space (X, τ ) is called a
pre-open set if A ⊆ int(cl(A)) and preclosed set if cl(int(A)) ⊆ A.
Definition 2.2 [5]: A subset A of a topological space (X, τ ) is called a
semi-open set if A ⊆ cl(int(A)) and semi closed set if int(cl(A)) ⊆ A.
Definition 2.3 [9]:A subset A of a topological space (X, τ ) is called an
α-open set if A ⊆ int(cl(int(A))) and an α -closed set if cl(int(cl(A))) ⊆ A.
Definition 2.4 [1]:A subset A of a topological space (X, τ ) is called a
semi-preopen set (β-open set) if A ⊆ cl(int(cl(A))) and semi-preclosed set
if int(cl(int(A))) ⊆ A.
Definition 2.5 [2]:A subset A of a topological space (X, τ ) is called a
b-open set if A ⊆ cl(int(A)) ∪ int(cl(A)) and b-closed set if cl(int(A)) ∪
int(cl(A)) ⊆ A.
Definition 2.6 [4]: A subset A of a topological space (X, τ ) is called a
generalized closed set(briefly g-closed) if cl(A) ⊆ U , whenever A ⊆ U and U
is open in X.
Definition 2.7 [6]: A subset A of a topological space (X, τ ) is called a
generalized α -closed (briefly gα -closed) if αcl(A) ⊆ U whenever A ⊆ U and
U is α-open in (X, τ ).
Definition 2.8 [10]: A subset A of a topological space (X, τ ) is called a
generalized b-closed set(briefly gb-closed) if bcl(A) ⊆ U , whenever A ⊆ U and
U is open in X.
Definition 2.9 [8]: A subset A of a topological space (X, τ ) is called a
weakly generalized closed set(briefly wg-closed) if cl(int(A)) ⊆ U , whenever
A ⊆ U and U is open in X.
g ∗ b-closed sets in topological spaces
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Definition 2.10 [3]: A subset A of a topological space (X, τ ) is called a
generalized semi-preclosed set(briefly gsp-closed) if spcl(A) ⊆ U , whenever
A ⊆ U and U is open in X.
Definition 2.11 [11]: A subset A of a topological space (X, τ ) is called a
generalized* closed set(briefly g ∗ -closed) if cl(A) ⊆ U , whenever A ⊆ U and U
is g-open in X.
3
Basic properties of g ∗b-closed sets
In this section we introduce the concept of g ∗ b-closed sets in topological space
and we investigate the group of structure of the set of all g ∗ b-closed sets.
Definition 3.1 Let (X, τ ) be a topological space and A be its subset, then
A is g ∗ b-closed set if bcl(A) ⊆ U whenever A ⊆ U and U is g-open in X.
Theorem 3.2 Every closed set is g ∗ b-closed set.
Proof:Let A be a closed set in X such that A ⊆ U , where U is g-open. Since
A is closed, cl(A) = A. Since bcl(A) ⊆ cl(A) = A. Therefore bcl(A) ⊆ U .
Hence A is a g ∗ b-closed set in X.
The converse of the above theorem need not be true as seen from the following
example.
Example 3.3 Let X={a,b,c} with the topology τ = {X, Φ, {a}, {a, b}}. Let
A = {a, c}. Here A is a g ∗ b-closed set but not a closed set of (X, τ ).
Theorem 3.4 Every b-closed set is g ∗ b-closed set.
Proof:Let A be a b-closed set in X such that A ⊆ U , where U is g-open. Since
A is b-closed, bcl(A) = A. Therefore bcl(A) ⊆ U . Hence A is a g ∗ b-closed set
in X.
The converse of the above theorem need not be true as seen from the following
example.
Example 3.5 Let X={a,b,c} with the topology τ = {X, Φ, {a}, {a, c}}. Let
A = {a, b}. Here A is a g ∗ b closed set but not a b-closed set of (X, τ ).
Theorem 3.6 Every g ∗ b-closed set is gb-closed set.
Proof: Let A be a g ∗ b-closed set in X such that A ⊆ U , where U is open. Since
every open set is g-open and g ∗ b-closed, bcl(A) ⊆ U . Hence A is gb-closed.
The converse of the above theorem need not be true as seen from the following
example.
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Example 3.7 Let X={a,b,c} with the topology τ = {X, φ, {a}}. Let A =
{a, c}. Here A is a gb-closed set but not a g ∗ b-closed set of (X, τ ).
Theorem 3.8 Every α-closed set is g ∗ b-closed set .
Proof: Let A be a α-closed set in X such that A ⊆ U , where U is g-open.
Since A is α-closed, bcl(A) ⊆ αcl(A) ⊆ U . Therefore bcl(A) ⊆ U . Hence A is
g ∗ b-closed.
The converse of the above theorem need not be true as seen from the following
example.
Example 3.9 Let X={a,b,c} with the topology τ = {X, Φ, {a}, {a, b}}. Let
A = {a, c}. Here A is a g ∗ b-closed set but not a α-closed set of (X, τ ).
Theorem 3.10 Every semi-closed set is g ∗ b-closed set .
Proof: Let A be a semi-closed set in X such that A ⊆ U , where U is g-open.
Since A is semi-closed, bcl(A) ⊆ scl(A) ⊆ U . Therefore bcl(A) ⊆ U . Hence A
is a g ∗ b-closed set in X.
The converse of the above theorem need not be true as seen from the following
example.
Example 3.11 Let X={a,b,c} with the topology τ = {X, Φ, {a, b}}. Let
A = {a}. Here A is a g ∗ b-closed set but not a semi-closed set of (X, τ ).
Theorem 3.12 Every preclosed set is g ∗ b-closed set .
Proof: Let A be a preclosed set in X such that A ⊆ U , where U is g-open.
Since A is preclosed, bcl(A) ⊆ pcl(A) ⊆ U . Therefore bcl(A) ⊆ U . Hence A is
g ∗ b-closed set in X.
The converse of the above theorem need not be true as seen from the following
example.
Example 3.13 Let X={a,b,c} with the topology τ = {X, Φ, {a}, {a, c}}.
Let A = {a, b}. Here A is a g ∗ b-closed set but not a pre-closed set of (X, τ ).
Theorem 3.14 Every g ∗ -closed set is g ∗ b-closed set .
Proof: Let A be a g ∗ -closed set in X such that A ⊆ U , where U is g-open.
Since A is g ∗ -closed, bcl(A) ⊆ cl(A) ⊆ U . Therefore bcl(A) ⊆ U . Hence A is a
g ∗ b-closed set in X.
The converse of the above theorem need not be true as seen from the following
example.
Example 3.15 Let X={a,b,c} with the topology τ = {X, Φ, {a}}. Let A =
{c}. Here A is a g ∗ b-closed set but not a g ∗ -closed set of (X, τ ).
g ∗ b-closed sets in topological spaces
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Theorem 3.16 Every gα-closed set is g ∗ b-closed set .
Proof: Let A be a gα-closed set in X such that A ⊆ U , where U is g-open.
Since A is gα-closed, bcl(A) ⊆ (A) ⊆ U . Therefore bcl(A) ⊆ U . Hence A is a
g ∗ b-closed set in X.
The converse of the above theorem need not be true as seen from the following
example.
Example 3.17 Let X={a,b,c} with the topology τ = {X, Φ, {a}, {a, b}}.
Let A = {a, c}. Here A is a g ∗ b closed set but not a gα-closed.
Theorem 3.18 Every g-closed set is g ∗ b-closed set.
Proof: Let A be a g-closed set in X such that A ⊆ U , where U is g-open.
Since A is g-closed, bcl(A) ⊆ cl(A) ⊆ U . Hence A is g ∗ b-closed.
The converse of the above theorem need not be true as seen from the following
example.
Example 3.19 Let X={a,b,c} with the topology τ = {X, Φ, {a}, {a, c}}.
Let A = {c}. Here A is a g ∗ b closed set but not a g-closed set of (x, τ ).
Theorem 3.20 Every g ∗ b-closed set is gsp-closed set.
Proof: Let A be a g ∗ b-closed set in X such that A ⊆ U , where U is open.
Since every open set is g-open and g ∗ b-closed , spcl(A) ⊆ bcl(A) ⊆ U . Hence
A is gsp-closed.
The converse of the above theorem need not be true as seen from the following
example.
Example 3.21 Let X={a,b,c} with the topology τ = {X, φ, {a}, {a, c}}.
Let A = {a, c}. Here A is a gsp-closed set but not a g ∗ b-closed set of (X, τ ).
Remark:We have the following implications but none of this implications are
reversible.
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closed
α-closed
g-closed
semi-closed
gsp-closed
b-closed
gb-closed
g*b-closed
preclosed
wg-closed
g*-closed
gα-closed
Theorem 3.22 A set A is g ∗ b-closed iff bcl(A) − A contains no non-empty
g-closed set.
Proof: Necessity: Let F be a g-closed set of (X, τ ) such that F ⊆ bcl(A)−A.
Then A ⊆ X − F . Since A is g ∗ b-closed and X − F is g-open then bcl(A) ⊆
X − F . This implies F ⊆ X − bcl(A). So F ⊆ (X − bcl(A)) ∩ (bcl(A) − A) ⊆
(X − bcl(A)) ∩ bcl(A) = φ. Therefore F = φ.
Sufficiency: Assume that bcl(A) − A contains no non-empty g-closed set. Let
A ⊆ U , U is g-open. Suppose that bcl(A) is not contained in U, bcl(A) ∩ U c
is a non-empty g-closed set of bcl(A) − A which is a contradiction. Therefore
bcl(A) ⊆ U and hence A is g ∗ b-closed.
Theorem 3.23 A g ∗ b-closed set A is b-closed if and only if bcl(A) − A is
b-closed.
Proof: If A is b-closed, then bcl(A) − A = φ. Conversely, suppose bcl(A) − A
is b-closed in X. Since A is g ∗ b-closed, by theorem 3.25, bcl(A) − A contains no
non-empty g-closed set in X. Then bcl(A) − A = φ. Hence A is b-closed.
g ∗ b-closed sets in topological spaces
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Theorem 3.24 If A and B are g ∗ b-closed, then A ∩ B is g ∗ b-closed.
Proof: Given that A and B are two g ∗ b-closed sets in X. Let A ∩ B ⊆ U , U
is g-open set in X. Since A is g ∗ b-closed, bcl(A) ⊆ U , whenever A ⊆ U , U is gopen in X. Since B is g ∗ b-closed, bcl(A) ⊆ U whenever B ⊆ U , U is g-open in X.
Corollary 3.25 The intersection of a g ∗ b-closed set and a closed set is a
g b-closed set.
∗
Remark:If A and B are g ∗ b-closed, then their union need not be g ∗ b-closed.
Example 3.26 The above remark is proved by the following example.
Let X={a,b,c} with the topology τ = {X, φ, {a}, {b}, {a, b}}. In this topological
space the subsets A = {a}, B = {b} are g ∗ b-closed but their union A ∪ B =
{a, b} is not g ∗ b-closed.
Theorem 3.27 If A is both g-open and g ∗ b-closed set of X, then A is bclosed.
Proof: Since A is g-open and g ∗ b-closed in X, bcl(A) ⊆ U . But always
A ⊆ bcl(A). Therefore A = bcl(A). Hence A is b-closed.
Theorem 3.28 For x ∈ X, the set X − {x} is g ∗ b-closed or g-open.
Proof: Suppose X − {x} is not g-open, then X is the only g-open set containing X − {x}. This implies bcl(X − {x}) ⊆ X. Then X − {x} is g ∗ b-closed in
X.
Theorem 3.29 If A is g ∗ b-closed and A ⊆ B ⊆ bcl(A), then B is g ∗ bclosed.
Proof: Let U be a g-open set of X such that B ⊆ U . Then A ⊆ U . Since
A is g ∗ b-closed, then bcl(A) ⊆ U . Now bcl(B) ⊆ bcl(bcl(A)) = bcl(A) ⊆ U .
Therefore B is g ∗ b-closed in X.
Theorem 3.30 Let A ⊆ Y ⊆ X and suppose that A is g ∗ b-closed in X,
then A is g ∗ b-closed relative to Y.
Proof: Given that A ⊆ Y ⊆ X and A is g ∗ b-closed in X. To show that A is
g ∗ b-closed relative to Y. Let A ⊆ Y ∩ U , where U is g-open in X. Since A is
g ∗ b-closed, A ⊆ U , implies bcl(A) ⊆ U . It follows that Y ∩ bcl(A) ⊆ Y ∩ U .
Thus A is g ∗ b-closed relative to Y.
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Theorem 3.31 Suppose that B ⊆ A ⊆ X, B is g ∗ b-closed set relative to A
and that A is both g-open and g ∗ b-closed subset of X, then B is g ∗ b-closed set
relative to X.
Proof: Let B ⊆ G and G be an open set in X. But given that B ⊆ A ⊆ X,
therefore B ⊆ A and B ⊆ G. This implies B ⊆ A ∩ G. Since B is g ∗ bclosed relative to A, A ∩ bcl(B) ⊆ A ∩ G. Implies (A ∩ bcl(B)) ⊆ G. Thus
(A ∩ bcl(B)) ∪ (bcl(B))c ⊆ G ∪ (bcl(B))c . Implies A ∪ (bcl(B))c ⊆ G ∪ (bcl(B))c .
Since A is g ∗ b-closed in X, we have bcl(A) ⊆ G ∪ (bcl(B))c . Also B ⊆ A
implies bcl(B) ⊆ bcl(A). Thus bcl(B) ⊆ bcl(A) ⊆ G ∪ (bcl(B))c . Therefore
bcl(B) ⊆ G, since bcl(B) is not contained in bcl(B)c . Thus B is g ∗ b-closed set
relative to X.
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sets,