Download closed sets in topological spaces

International Journal of Mathematical Archive-4(3), 2013, 309-315
Available online through www.ijma.info ISSN 2229 – 5046
ON µ̂ -CLOSED SETS IN TOPOLOGICAL SPACES
S. Pious Missier1 & E. Sucila2*
1
Department of Mathematics, V. O. Chidambaram College, Tuticorin – 628 008, India
2
Department of Mathematics, G. Venkataswamy Naidu College, Kovilpatti, India
(Received on: 06-01-13; Revised & Accepted on: 14-02-13)
ABSTRACT
In this paper, we introduce a new class of sets namely, µ̂ -closed sets and their properties. Applying these sets, we
introduce six new spaces namely, T µ̂ , αT µ̂ , sT µ̂ , pT µ̂ , spT µ̂ and µT µ̂ -spaces.
Keywords:
µ̂ -closed set, µ̂ -open set, T µ̂ , αT µ̂ , sT µ̂ , pT µ̂ , spT µ̂
and µT µ̂ - Spaces.
2010 Mathematics Subject Classification: 54A05, 54A10, 54D10.
1. INTRODUCTION
N. Levine [4] introduced the class of g–closed sets in 1970. Andrijevic [1], N. Levine [4] Mashoor et al [6] have
respectively introduced semipreclosed sets, semiclosed sets, preclosed sets which are some weak forms of closed sets.
M. K. R. S. Veerakumar has introduced several generalized closed sets namely, g*-closed sets, *g–closed sets, α*g–
closed sets, *gs–closed sets, ĝ -closed sets presemiclosed sets, µ-closed sets, µs–closed sets and µp–closed sets. In this
paper we introduce
µ̂ -closed
sets and applying these sets six spaces namely T µ̂ ,αT µ̂ , sT µ̂ , pT µ̂ , spT µ̂ and
µT µ̂ - spaces are introduced.
2. PRELIMINARIES
Throughout this paper, we consider spaces on which no separation axioms are assumed unless explicity stated. For A ⊂
X, the closure and interior of A is denoted by cl(A) and int(A) respectively. The complement of A is denoted by AC, the
power set of X is denoted by P(X).
Definition 2.1: A subset A of a topological space (X, τ) is called
1. a preopen set [6] if A ⊆ int(cl(A)) and preclosed if cl(int(A)) ⊆ A.
2. a semiopen set [3] if A ⊆ cl(int(A)) and a semiclosed set if int(cl(A)) ⊆ A.
3. an α-open set [7] if A ⊆ int(cl(int(A)) and α-closed set if cl(int(cl(A))) ⊆ A.
4. a semipreopen set [1] if A ⊆ cl(int(cl(A))) and a semipreclosed set if (cl(int(A))) ⊆ A.
The intersection of all semiclosed (resp. preclosed, semipreclosed, α-closed) sets containing a subset A of X is called
semiclosure (resp. preclosure, semipreclosure, α-closure ) of A is denoted by scl(A) (resp. pcl(A), spcl(A), αcl(A)).
The union of all semiopen sets containted in A is called semiinterior of A and is denoted by sint (A).
Definition 2.2: A subset A of a topological space (X, τ) is called
1. a generalized closed set (briefly g–closed [4] if cl(A) ⊆ U whenever A ⊆ U and U is open in (X, τ).
2. an α-generalized closed set (briefly αg–closed ) [6] if αcl(A) ⊆ U whenever A ⊆ U and U is open in (X, τ).
3. a semi generalized closed set (briefly sg–closed )[2] if scl(A) ⊆ U whenever A ⊆ U and U is semiopen in (X, τ).
4. a ĝ -closed set [11] if cl(A) ⊆ U whenever A ⊆ U and U is semiopen in (X, τ).
5. a *g–closed set [12] if cl(A) ⊆ U whenever A ⊆ U and U is ĝ -open in (X, τ).
6. a g*-closed set [12] if cl(A) ⊆ U whenever A ⊆ U and U is g-open in (X, τ).
2
Corresponding author: E. Sucila2*
Department of Mathematics, G. Venkataswamy Naidu College, Kovilpatti, India
International Journal of Mathematical Archive- 4(3), March – 2013
309
1
2*
S. Pious Missier & E. Sucila /ON
µ̂ -CLOSED SETS IN TOPOLOGICAL SPACES/ IJMA- 4(3), March.-2013.
7. a g*-preclosed set (briefly g*p–closed ) [13] if pcl(A) ⊆ U whenever A ⊆ U and U is g–open in (X, τ).
8. a *g- semiclosed set [17] (briefly *gs-closed ) if scl(A) ⊆ U whenever A ⊆ U and U is ĝ -open in (X, τ).
9. a α*g-closed set [17] if αcl(A) ⊆ U whenever A ⊆ U, and U is ĝ -open in (X, τ).
10. a gα*-closed set [5] if αcl(A) ⊆ int (U) whenever A ⊆ U and U is α-open in (X, τ).
11. a ψ-closed set [15] if scl(A) ⊆ U whenever A ⊆ U and U is sg–open in (X, τ).
12. a g* ψ-closed set [15] if ψ cl(A) ⊆ U whenever A ⊆ U and U is g–open in (X, τ).
13. a µ-closed set [16] if cl(A) ⊆ U whenever A ⊆ U and U is gα*-open in (X, τ).
14. a µ-preclosed set (briefly µp–closed ) [17] if pcl(A) ⊆ U whenever A ⊆ U and U is gα*- open in (X, τ) .
15. a µ-semiclosed set (briefly µs–closed) [18] if scl(A) ⊆ U whenever A ⊆ U and U is gα*-open in (X, τ).
Notations 2.3
1. αC(X, τ) is the class of α-closed subsets of (X, τ).
2. sC(X, τ) is the class of semiclosed subsets of (X, τ).
3. pC(X, τ) is the class of preclosed subsets of (X, τ).
4. spC(X, τ) is the class of semipreclosed subsets of (X, τ).
3. PROPERTIES OF µ̂ -CLOSED SETS
We introduce the following definition.
µ̂ -closed set if scl(A) ⊆ U whenever A ⊆ U and U is
The class of µ̂ -closed subsets of X is denoted by µ̂ C(X,τ).
Definition 3.1: A subset A of (X, τ) is called
Proposition 3.2: Every closed set (resp. α-closed set, semiclosed set) is
can be seen from the following examples.
Example 3.3: Let X = {a, b, c} and τ = {x, ϕ, {c}, {a, b}}. Hence
Here the set {a} is
µ-open in (X, τ).
µ̂ -closed. But the converses are not true as
µ̂ C(X, τ) = P(X).
µ̂ -closed but it is not closed (resp. not semi closed, not α-closed).
Thus the class of
µ̂ -closed sets properly contains the classes of closed sets, α-closed sets, semiclosed sets.
Proposition 3.4:
µ̂ -closedness is independent of g-closedness and αg-closedness.
Proof: It follows from the following examples.
Example 3.5: Let X = {a, b, c}, τ = {X, ϕ, {a}, {a, b}}. Here the set {a, c} is g–closed and αg–closed but it is not
closed.
Example 3.6: Let X = {a, b, c}, τ = {X,ϕ, {a}, {b}, {a, b}}. Here the set {a}
αg–closed.
Proposition 3.7:
µ̂ -
µ̂ - closed but it is neither g–closed nor
µ̂ -closedness is independent of *g-closedness, α*g-closedness and *gs– closedness.
Proof: It follows from the following examples.
Example 3.8: Let X = {a, b, c}, τ = {X,ϕ, {a}, {a, b}}. Here the set {a, c} is *g–closed, α*g-closed and *gs–closed but
it is not µ̂ -closed.
Example 3.9: Let X = {a, b, c}, τ = {X, ϕ, {a}, {b, c}}. Here the set {b} is
α*g-closed and it is not even *gs–closed.
Preposition 3.10:
µ̂ -closed but it is neither *g- closed nor
µ̂ -closedness is independent of g*-closedness and g*p–closedness.
Proof: It follows from the following examples.
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µ̂ -CLOSED SETS IN TOPOLOGICAL SPACES/ IJMA- 4(3), March.-2013.
Example 3.11: Let X = {a, b, c}, τ = {X, ϕ, {a}, {a, c}}. Here the set {a, b} is g*-closed and g*p–closed but it is not
µ̂ -closed.
Example 3.12: Let X = {a, b, c}, τ = {X, ϕ, {b}, {c}, {b, c}}. Here the set {b} is
nor g*p–closed.
Proposition 3.13:
µ̂ -closed but it is neither g*-closed
µ̂ -closedness is independent of g*ψ-closedness.
Proof: It follows from the following examples.
Example 3.14: Let X = {a, b, c}, τ = {X, ϕ, {a}, {a, b}}. Here the set {c, a} g*ψ-closed but it is not
Example 3.15: Let X = {a, b, c}, τ = {X, ϕ, {a}, {b, c}}. Here the set {b} is
Proposition 3.16:
µ̂ -closed.
µ̂ -closed but it is not g* ψ-closed.
µ̂ -closedness is independent of µ-closedness.
Proof: It follows from the following examples.
Example 3.17: Let X = {a, b, c}, τ = {X, ϕ, {c}, {a, b}}. Here the set {a} is
µ̂ -closed but it is not
Example 3.18: Let X = {a, b, c}, τ = {X, ϕ, {c}, {a, c}}. Here the set {a, c} is µ-closed but it is not
Proposition 3.19:
µ-closed.
µ̂ -closed.
µ̂ -closedness is independent of µp–closedness and µs-closedness.
Proof: It follows from the following examples.
Example 3.20: Let X = {a, b, c}, τ = {X, ϕ, {a}, {a, b}}. Here the set {a, b} is µp–closed and µs–closed but it is not
µ̂ -closed.
Example 3.21: Let X = {a, b, c}, τ = {X,ϕ, {a}, {c} {a, c}}. Here the set {a} is
Example 3.22: Let X = {a, b, c}, τ = {X,ϕ, {a}, {b, c}}. Here the set {b} is
Remark 3.23: The union (intersection) of any two
µ̂ -closed but it is not µp–closed.
µ̂ -closed but it is not µs–closed.
µ̂ -closed sets is not µ̂ -closed For,
Example 3.24: Let X and τ be defined as in Example 3.6. Here the sets {a} and {b} are
= {a, b} is not
µ̂ -closed sets, but {a} ∪ {b}
µ̂ -closed set.
Example 3.25: Let X = {a, b, c, d}, τ = {X, ϕ, {a, b}, {c}, {a, b, c}}. Here the sets {a, b} and {a, d} are
but {a, b} ∩ {a, d} = {a} it is not
µ̂ -closed sets,
µ̂ -closed set.
Proposition 3.26: Let A and B be any two subsets of the topology (X, τ). Then
1. A is µ̂ -closed, then scl(A) \ A does not contain any non empty µ-closed set.
2.
A is
µ̂ -closed and A ⊂ B ⊂ scl(A), then B is µ̂ -closed.
µ̂ -closed and suppose scl(A) \ A contain a µ-closed set F. Therefore F ⊂ scl(A) \ A implies A ⊂ FC,
which is µ-open. Since A is µ̂ -closed, scl(A) ⊂ FC implies F ⊂ (scl(A))C, also F ⊂ scl(A) therefore F ⊂ scl(A) ∩
Proof: Let A be
(scl(A))C = ϕ.
Let U be a µ-openset such that B ⊂ U. Since A ⊂ B ⊂ U and U is µ-open scl(A) ⊂ U. Since B ⊂ scl(A) , scl(B) ⊂
scl(scl(A)) implies scl(B) ⊂ scl(A) ⊂ U therefore B is µ̂ -closed.
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µ̂ -CLOSED SETS IN TOPOLOGICAL SPACES/ IJMA- 4(3), March.-2013.
Remark 3.27: The following diagram shows the relationship established between µ̂ -closed set and some other sets
A
B (resp. A
B) represents A implies B but not conversely (resp. A and B are independent of each other).
From the above Propositions and Examples, we have the following diagram.
µs-closed
µ-closed
µp-closed
|
g*p-closed
closed
g*-closed
g- closed
µ̂ -closed
α-closed
αg-closed
semiclosed
g*ψ-closed
α*g-closed
*g-closed
*gs-closed
Diagram: I
Definition 3.28: A subset A of a space X is said to be
is denoted by
µ̂ -open if AC is µ̂ -closed. The class of all µ̂ -open subsets of X
µ̂ O(X, τ).
Proposition 3.29: A subset A of a topological space X is said to be
and F is µ-closed in X.
µ̂ -open if and only if F ⊂ sint(A) whenever A ⊃ F
Proof: Suppose that A is µ̂ -open in X and A ⊃ F, where F is µ-closed in X. Then AC ⊂ FC, where FC is µ-open in X.
Hence we get scl(AC) ⊂ FC implies (sint(A))C⊂ FC. Thus, we have sint(A) ⊃ F.
Conversely, suppose that AC ⊂ U and U is µ-open in X then A ⊃ UC and UC is µ-closed then by hypothesis sint(A) ⊃
UC implies (sint(A)) C ⊂ U. Hence scl(AC) ⊂ U gives AC is µ̂ -closed.
Proposition 3.30: In a topological space X, for each x ∈ X, either {x} is µ-closed or
µ̂ -open in X.
Proof: Suppose that {x} is not µ-closed in X, then X – {x} is not µ-open and the only µ-open set containing X–{x} is
the space X itself. Therefore, scl(X – {x}) ⊂ X and so X–{x} is µ̂ -closed gives {x} is µ̂ -open.
4. APPLICATION OF
µ̂ -CLOSED SETS
As applications of µ̂ -closed sets, new spaces namely, T µ̂ , αT µ̂ , sT µ̂ , pT µ̂ , spT µ̂ and µT µ̂ spaces are
introduced. First we introduce the following definitions.
Definition 4.1: A topological space (X, τ) is called a
1. T µ̂ -space if every µ̂ -closed set is closed.
2. αT µ̂ -space if every
µ̂ -closed set is α closed.
3. sT µ̂ -space if every µ̂ -closed set is semiclosed.
4. pT µ̂ -space if every µ̂ -closed set is preclosed.
5. spT µ̂ -space if every µ̂ -closed set is semipreclosed.
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Example 4.2: Let X = {a, b, c} and τ = {X, ϕ, {a}, {a, b}, {a, c}}. Here
µ̂ C(X, τ) = {X, ϕ, {b}, {c}, {b, c}}. Then
(X, τ) is a T µ̂ -space. The space in the following example is not a T µ̂ -space. Let X = {a, b, c} and τ = {X, ϕ, {c}, {a,
b}}. Here µC(X, τ) = P(X).
µ̂ C(X, τ) = {X, ϕ, {b}, {c}, {b, c}} = αC(X, τ).
Thus (X, τ) is a αT µ̂ -space. The space in the following example is not a αT µ̂ -space. Let X = {a, b, c} and τ = {X, ϕ,
{b}, {c, a}}. Here µ̂ C(X, τ) = P(X) and αC(X, τ) = {X, ϕ, {b}, {a, c}}. Therefore (X, τ) is not a αT µ̂ -space.
Example 4.3: Let X = {a, b, c} and τ = {X, ϕ, {a}, {a, c}}. Here
Proposition 4.4: If (X, τ) is a αT µ̂ -space then every singleton of X is either µ-closed or semiopen.
Proof: Let x ∈ X. Suppose {x} is not µ-closed, then X–{x} is not µ-open. This implies that X is the only µ-open set
containing X–{x}. So X–{x} is µ̂ -closed of (X, τ). Since (X, τ) is αT µ̂ -closed, X–{x} is α-closed and every α-closed
is semiclosed implies X–{x} is semiclosed or equivalently {x} is semiopen. The converse of the above proposition is
not true as it can be seen by the following example.
Example 4.5: Let X = {a, b, c} and τ = {X, ϕ, {a}, {b}, {a, b}}. Here every singleton of X is either µ-closed or semi
open but (X, τ) is not αT µ̂ -space.
Proposition 4.6: Every αT µ̂ -space is sT µ̂ (resp. pT µ̂ )-space.
Proof: It follows from the fact that every α-closed is semi (resp. pre) closed. The converse of the above proposition is
not true as it can be seen by the following example.
Example 4.7: Let X = {a, b, c} and τ = {X, ϕ, {b}, {c}, {b, c}}. Here (X, τ) is sT µ̂ -space but it is not a αT µ̂ -space.
Example 4.8: Let X = {a, b, c} and τ = {X, ϕ, {a}, {b,c}}. Here (X, τ) is pT µ̂ -space but it not a αT µ̂ -space.
µ̂ C(X, τ) = {X, ϕ, {b}, {c}, {b, c}} = αC(X, τ) =
pC(X, τ) = sC(X, τ) = spC(X, τ). Therefore here the space (X, τ) is αT µ̂ -space, pT µ̂ -space, sT µ̂ -space, spT µ̂ -
Example 4.9: Let X = {a, b, c} and τ = {X, ϕ, {a}, {a, b}}. Here
space.
Proposition 4.10: Every T µ̂ -space is αT µ̂ -space, pT µ̂ -space, sT µ̂ -space and spT µ̂ -space but not conversely.
Example 4.11:
The space (X, τ) in Example 4.9 is αT µ̂ -space, pT µ̂ -space, sT µ̂ -space and spT µ̂ -space but not
T µ̂ -space.
Definition 4.12: A space (X, τ) is called µT µ̂ -space if every
µ̂ -closed set is µ-closed.
Example 4.13: Let X = {a, b, c} and τ = {X, ϕ, {b, c}}. Here
µ̂ C(X, τ) = {X, ϕ, {a}, {a, b}, {c, a}} = µC(X, τ).
Therefore (X, τ) is a µT µ̂ -space.
Example 4.14: Let X = {a, b, c} and τ = {X, ϕ, {c}, {a, b}}. Here
µ̂ C(X, τ) = P(X) and µC(X, τ) = {X, ϕ, {c}, {a,
b}}. Therefore (X, τ) is not a µT µ̂ -space.
Proposition 4.15: Every T µ̂ (resp. αT µ̂ ) space is µT µ̂ -space, but not conversely.
µ̂ -closed set in a topological space X, which is T µ̂ -space. Hence A is closed implies A is µ-closed.
Therefore T µ̂ -space is µT µ̂ -space. Similarly A is µ̂ -closed set in topological space X which is αT µ̂ -space. Hence
Proof: Let A be
A is α-closed implies A is µ-closed.
Therefore αT µ̂ -space is µT µ̂ -space. Hence the proof.
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µ̂ -CLOSED SETS IN TOPOLOGICAL SPACES/ IJMA- 4(3), March.-2013.
Converse is not true as it can be seen by the following example. The space (X, τ) in Example 4.13 is µT µ̂ -space but it
is neither T µ̂ -space nor αT µ̂ -space.
Proposition 4.16: If X is a µT µ̂ -space then every singleton of X is either µ-closed or µ-open.
Proof: By Proposition 3.30 and by Definition 4.12, we get the proof.
Remark 4.17: The converse of the Proposition 4.16 is not true as it can be seen by the following example. Let X = {a,
b, c} and τ = {X, ϕ, {a}, {b}, {a,b}}. Here the space (X, τ) satisfies the condition of above Proposition 4.16 but (X, τ)
is not µT µ̂ -space.
From the above Propositions and Examples, we have the following diagram.
pT µ̂
spT µ̂
T µ̂
αT µ̂
sT µ̂
µT µ̂
Diagram: II
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Source of support: Nil, Conflict of interest: None Declared
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