Download Mohammad S. Sarsak ON SOME PROPERTIES OF GENERALIZED

DEMONSTRATIO MATHEMATICA
Vol. XLVI
No 2
2013
Mohammad S. Sarsak
ON SOME PROPERTIES OF GENERALIZED OPEN SETS
IN GENERALIZED TOPOLOGICAL SPACES
Abstract. In [8], Császár introduced some generalized open sets in generalized topological spaces, namely, µ-semi-open, µ-preopen, µ-β-open, and µ-α-open. The primary
purpose of this paper is to study some new properties and characterizations of these concepts as we also introduce and study a new generalized open set in generalized topological
spaces, namely, µ-b-open. We also investigate some relationships between these concepts.
1. Introduction and preliminaries
A generalized topology (breifly GT) [7] µ on a nonempty set X is a collection of subsets of X such that ∅ ∈ µ and µ is closed under arbitrary unions.
Elements of µ will be called µ-open sets, and a subset A of (X, µ) will be
called µ-closed if X\A is µ-open. Clearly, a subset A of (X, µ) is µ-open if
and only if for each x ∈ A, there exists Ux ∈ µ such that x ∈ Ux ⊂ A, or
equivalently, A is the union of µ-open sets. The pair (X, µ) will be called
generalized topological space (breifly GTS). By a space X or (X, µ), we
will always mean a GTS. A space (X, µ) is called a µ-space [16] if X ∈ µ.
(X, µ) is called a quasi-topological space [9] if µ is closed under finite intersections. Clearly, every topological space is a quasi-topological space, every
quasi-topological space is a GTS, and a space (X, µ) is a topological space
if and only if (X, µ) is both µ-space and quasi-topological space.
If A is a subset of a space (X, µ), then the µ-closure of A [8], cµ (A), is
the intersection of all µ-closed sets containing A and the µ-interior of A [8],
iµ (A), is the union of all µ-open sets contained in A. It was pointed out in [8]
that each of the operators cµ and iµ are monotonic [10], i.e. if A ⊂ B ⊂ X
then cµ (A) ⊂ cµ (B) and iµ (A) ⊂ iµ (B), idempotent [10], i.e. if A ⊂ X
then cµ (cµ (A)) = cµ (A) and iµ (iµ (A)) = iµ (A), cµ is enlarging [10], i.e. if
2010 Mathematics Subject Classification: Primary 54A05, 54A10.
Key words and phrases: µ-open, µ-closed, µ-regular open, µ-regular closed, µ-semiopen, µ-dense, µ-preopen, µ-β-open, µ-α-open, µ-b-open, generalized topology.
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A ⊂ X then cµ (A) ⊃ A, iµ is restricting [10], i.e. if A ⊂ X then iµ (A) ⊂ A,
A is µ-open if and only if A = iµ (A), and cµ (A) = X\iµ (X\A).
Clearly, A is µ-closed if and only if A = cµ (A), cµ (A) is the smallest
µ-closed set containing A, iµ (A) is the largest µ-open set contained in A,
x ∈ cµ (A) if and only if any µ-open set containing x intersects A, and
x ∈ iµ (A) if and only if there exists a µ-open set U such that x ∈ U ⊂ A.
A subset A of a topological space X is called regular open if A = IntA,
and regular closed if X\A is regular open, or equivalently, if A = IntA.
A is called semi-open [13] (resp. preopen [14] where called locally dense
in [6], semi-preopen [3] where called β-open in [1], b-open [4], α-open [15]) if
A ⊂ IntA (resp. A ⊂ Int A, A ⊂ IntA, A ⊂ IntA ∪ Int A, A ⊂ Int IntA). A is
called semi-regular [11] if it is both semi-open and semi-closed, or equivalently, if there exists a regular open set U such that U ⊂ A ⊂ U , where A
and IntA denote respectively, the closure of A in X and the interior of A
in X. In [5], the term regular semi-open were used for a semi-regular set. For
the concepts and terminology not defined here, we refer the reader to [12].
In concluding this section, we recall the following definitions and facts
for their importance in the material of our paper.
Definition 1.1. [8] Let A be a subset of a space (X, µ). Then A is called
(i)
(ii)
(iii)
(iv)
µ-semi-open if A ⊂ cµ (iµ (A)).
µ-preopen if A ⊂ iµ (cµ (A)).
µ-β-open if A ⊂ cµ (iµ (cµ (A))).
µ-α-open if A ⊂ iµ (cµ (iµ (A))).
As in [8], for a space (X, µ), we will denote the class of µ-semi-open sets
by σ, the class of µ-preopen sets by π, the class of µ-β-open sets by β, and
the class of µ-α-open sets by α.
Proposition 1.2. [8] Let (X, µ) be a space. Then
(i) µ ⊂ α ⊂ σ ⊂ β.
(ii) α ⊂ π ⊂ β.
(iii) α = σ ∩ π.
Proposition 1.3. [8] Let (X, µ) be a space. Then each of the families σ,
π, α, and β is a generalized topological space.
Definition 1.4. [18] Let A be a subset of a space (X, µ). Then A is called
(i)
(ii)
(iii)
(iv)
µ-regular closed if A = cµ (iµ (A)).
µ-regular open if X\A is µ-regular closed.
µ-semi-closed if X\A is µ-semi-open.
µ-preclosed if X\A is µ-preopen.
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(v) µ-β-closed if X\A is µ-β-open.
(vi) µ-α-closed if X\A is µ-α-open.
We will denote the collection of all µ-regular open sets by r.
Proposition 1.5. [18] Let A be a subset of a space (X, µ). Then
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
A
A
A
A
A
A
A
is
is
is
is
is
is
is
µ-semi-closed if and only if iµ (cµ (A)) ⊂ A.
µ-preclosed if and only if cµ (iµ (A)) ⊂ A.
µ-regular open if and only if A = iµ (cµ (A)).
µ-β-closed if and only if iµ (cµ (iµ (A))) ⊂ A.
µ-α-closed if and only if cµ (iµ (cµ (A))) ⊂ A.
µ-regular open if and only if A = iµ (B) for some µ-closed set B.
µ-regular closed if and only if A = cµ (B) for some µ-open set B.
Proposition 1.6. [18] For a subset A of a space (X, µ), the following are
equivalent:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
A
A
A
A
A
A
is
is
is
is
is
is
µ-regular open,
µ-open and µ-semi-closed,
µ-open and µ-β-closed,
µ-α-open and µ-β-closed,
µ-α-open and µ-semi-closed,
µ-preopen and µ-semi-closed.
Corollary 1.7. [18] For a subset A of a space (X, µ), the following are
equivalent:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
A
A
A
A
A
A
is
is
is
is
is
is
µ-regular closed,
µ-closed and µ-semi-open,
µ-closed and µ-β-open,
µ-α-closed and µ-β-open,
µ-α-closed and µ-semi-open,
µ-preclosed and µ-semi-open.
2. µ-b-open sets
Definition 2.1. Let A be a subset of a space (X, µ). Then A is called
µ-b-open if A ⊂ cµ (iµ (A)) ∪ iµ (cµ (A)). A is called µ-b-closed if X\A is
µ-b-open, or equivalently, if cµ (iµ (A)) ∩ iµ (cµ (A)) ⊂ A. We will denote the
class of all µ-b-open subsets of (X, µ) by b.
Remark 2.2. It is easy to see that for a space (X, µ), σ ∪ π ⊂ b ⊂ β.
It was shown in [4] that in case of a topological space, the above inclusions
are not reversible.
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Remark 2.3. It is also easy to see using the monotonicity of cµ and iµ ,
that for a space (X, µ), (X, b) is a generalized topological space.
The following proposition is an immediate consequence of Remarks 2.2
and 2.3.
Proposition 2.4. Let A be a subset of a space (X, µ). If A = B ∪ C,
where A is µ-semi-open and C is µ-preopen, then A is µ-b-open.
In [17], it is pointed out that a subset A of a topological space X is
b-open if and only if A = B ∪ C, where B is semi-open and C is preopen.
However, this is not true for generalized topological spaces since the converse
of Proposition 2.4 need not be true as following example shows.
Example 2.5. Let
B = {{4} , {1, 2, 3} , {3, 4, 5} , {5, 6, 7}}
and let
µ = {∅, all possible unions of members of B} .
Then (X, µ) is a generalized topological space, where X = {1, 2, 3, 4, 5, 6, 7}.
If A = {1, 2, 4}. Then A is µ-b-open because
cµ (A) = {1, 2, 3, 4} , and thus iµ (cµ (A)) = {1, 2, 3}
and
iµ (A) = {4} , and thus cµ (iµ (A)) = {4}.
We will show now that there is no µ-semi-open set B and µ-preopen set
C such that A = B ∪ C. Observe first that the only µ-semi-open subsets of
A are ∅ and {4}, that is because cµ (iµ (A)) = {4}. We discuss the following
cases:
(i) If B = ∅, then C = {1, 2, 4} which is not µ-preopen because cµ (U ) =
{1, 2, 3, 4} and iµ (cµ (C)) = {1, 2, 3}.
(ii) If B = {4} and C = {1, 2, 4}, then C is not µ-preopen.
(iii) If B = {4} and C = {1, 2}, then C is not µ-preopen because cµ (C) =
{1, 2} and iµ (cµ (C)) = ∅.
The following result is an immediate consequence of Proposition 1.6 and
Remark 2.2.
Corollary 2.6. For a subset A of a space (X, µ), the following are equivalent:
(i) A is µ-regular open,
(ii) A is µ-open and µ-b-closed,
(iii) A is µ-α-open and µ-b-closed.
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Lemma 2.7. Let A be a subset of a space (X, µ). If A is both µ-semi-closed
and µ-β-open, then A is µ-semi-open.
Proof. Since A is µ-semi-closed, it follows from Proposition 1.5(i) that
iµ (cµ (A)) ⊂ A.
But A is µ-β-open, so
iµ (cµ (A)) ⊂ A ⊂ cµ (iµ (cµ (A))) .
Thus
iµ (cµ (A)) ⊂ iµ (A) .
Therefore
A ⊂ cµ (iµ (cµ (A))) ⊂ cµ (iµ (A)) .
Hence, A is µ-semi-open.
Proposition 2.8. Let A be a µ-β-open subset of a space (X, µ). Then
cµ (A) is µ-regular closed.
Proof. Since A is µ-β-open, A ⊂ cµ (iµ (cµ (A))). Thus
cµ (A) ⊂ cµ (iµ (cµ (A))) ⊂ cµ (A) .
Therefore, cµ (A) = cµ (iµ (cµ (A))). Hence, by Proposition 1.5(vii), cµ (A)
is µ-regular closed.
Corollary 2.9. Let A be a subset of a space (X, µ). If A ∈ b (resp.
A ∈ σ, A ∈ π, A ∈ α), then cµ (A) is µ-regular closed.
Definition 2.10. A subset A of a space (X, µ) is called regular µ-semiopen if there exists a µ-regular open set U such that U ⊂ A ⊂ cµ (A). We
will denote the collection of all regular µ-semi-open sets by rσ.
Remark 2.11. It is clear that every µ-regular open (µ-regular closed) set
is regular µ-semi-open. However, it is well known that the converse is not
true even for topological spaces.
Proposition 2.12. Let A be a subset of a space (X, µ). Then the following
are equivalent:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
A
A
A
A
A
A
is
is
is
is
is
is
regular µ-semi-open,
µ-semi-open and µ-semi-closed,
µ-b-open and µ-semi-closed,
µ-β-open and µ-semi-closed,
µ-semi-open and µ-b-closed,
µ-semi-open and µ-β-closed.
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Proof. (i)→(ii): Let U be a µ-regular open set such that U ⊂ A ⊂ cµ (A).
Then U ⊂ iµ (A) and therefore
A ⊂ cµ (U ) ⊂ cµ (iµ (A)).
Thus, A is µ-semi-open. On the other hand, since cµ (A) = cµ (U ) and U is
µ-regular open, it follows that
U = iµ (cµ (U )) = iµ (cµ (A)) ⊂ A.
Thus, by Proposition 1.5(i), A is µ-semi-closed.
(ii)→(iii), (iii)→(iv): Follow since σ ⊂ b ⊂ β.
(iv)→(v): Follows from Lemma 2.7 and since σ ⊂ b.
(v)→(vi): Follows since b ⊂ β.
(vi)→(i): Since A is µ-semi-open and µ-β-closed, it follows from Lemma 2.7 that A is µ-semi-closed. Thus, by Proposition 1.5(i)
iµ (cµ (A)) ⊂ A ⊂ cµ (iµ (A)) ⊂ cµ (iµ (cµ (A))).
Let U = iµ (cµ (A)). Then
U ⊂ A ⊂ cµ (A).
By Proposition 1.5(vi), U is µ-regular open. Hence, A is regular µ-semiopen.
Remark 2.13. It is clear from Proposition 2.12, that if A is a regular
µ-semi-open subset of a space (X, µ), then X\A is regular µ-semi-open too.
3. Various properties
Proposition 3.1. Let (X, µ) be a µ-space and let x ∈ X. Then {x} is
µ-open if and only if {x} is µ-semi-open.
Proof. The necessity is clear. Assume that {x} is µ-semi-open. Then {x} ⊂
cµ (iµ ({x})). Now iµ ({x}) is either {x} or ∅. Since (X, µ) is a µ-space,
cµ (∅) = ∅, but {x} ⊂ cµ (iµ ({x})), so iµ ({x}) 6= ∅. Therefore, iµ ({x}) =
{x}, that is, {x} is µ-open.
The following lemma can be easily verified.
Lemma 3.2. Let (X, µ) be a space, A ⊂ X, and U ∈ µ. If A ∩ U = ∅ then
cµ (A) ∩ U = ∅.
Proposition 3.3. Let (X, µ) be a µ-space and let x ∈ X. Then the
following are equivalent:
(i) {x} is µ-preopen,
(ii) {x} is µ-b-open,
(iii) {x} is µ-β-open.
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Proof. (i)→(ii) and (ii)→(iii) follows from Remark 2.2. Assume that {x}
is µ-β-open and that {x} is not µ-preopen. Then {x} 6⊂ iµ (cµ ({x})), that
is, {x} ∩ iµ (cµ ({x})) = ∅. Since iµ (cµ ({x})) is µ-open, it follows from
Lemma 3.2 that cµ ({x}) ∩ iµ (cµ ({x})) = ∅, and thus, iµ (cµ ({x})) = ∅.
Since (X, µ) is a µ-space, it follows that
cµ (iµ (cµ ({x}))) = cµ (∅) = ∅
but {x} is µ-β-open, a contradiction.
Proposition 3.4. Let (X, µ) be a µ-space and let x ∈ X. Then {x} is
µ-preopen or {x} is µ-α-closed.
Proof. Assume that {x} is not µ-preopen. Then {x} 6⊂ iµ (cµ ({x})), that
is, {x} ∩ iµ (cµ ({x})) = ∅. Since iµ (cµ ({x})) is µ-open, it follows from
Lemma 3.2 that cµ ({x}) ∩ iµ (cµ ({x})) = ∅, and thus, iµ (cµ ({x})) = ∅.
Since (X, µ) is a µ-space, it follows that cµ (iµ (cµ ({x}))) = cµ (∅) = ∅.
Thus by Proposition 1.5(v), {x} is µ-α-closed.
Definition 3.5. A subset A of a space (X, µ) is called µ-dense if cµ (A)
= X, or equivalently, if for every nonempty µ-open set U , U ∩ A 6= ∅. A is
called µ-codense if X\A is µ-dense, or equivalently, if iµ (A) = ∅.
It is well known, in topological spaces, that a µ-dense set is µ-preopen,
and that a µ-preopen set need not be µ-dense. The following example shows
that µ-dense sets in generalized topological spaces need not be µ-preopen.
Example 3.6. Let X = {a, b, c, d}, µ = {∅, {a, b} , {b, c} , {a, b, c}}. If
A = {b, d}, then clearly A is µ-dense. However, A is not µ-preopen since
cµ (A) = X and iµ (X) = {a, b, c}.
Remark 3.7. Let (X, µ) be a µ-space. If A is µ-dense, then A is µ-preopen.
Proposition 3.8. Let A be a subset of a space (X, µ). Then A is µ-semiopen if and only if there exists a µ-open set U such that U ⊂ A ⊂ cµ (U ).
Proof. Necessity. Suppose that A is µ-semi-open. Then A ⊂ cµ (iµ (A)).
Let U = iµ (A). Then U ⊂ A ⊂ cµ (U ).
Sufficiency. Assume that there exists a µ-open set U such that U ⊂ A ⊂
cµ (U ). Then U ⊂ iµ (A) and thus cµ (U ) ⊂ cµ (iµ (A)), but A ⊂ cµ (U ), so
A ⊂ cµ (iµ (A)), that is, A is µ-semi-open.
Proposition 3.9. Let A be a subset of a space (X, µ). If there exists a
µ-preopen set U such that U ⊂ A ⊂ cµ (U ), then A is µ-β-open.
Proof. Since U ⊂ A ⊂ cµ (U ), it follows that cµ (A) = cµ (U ), and thus
iµ (cµ (A)) = iµ (cµ (U )) ,
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M. S. Sarsak
but U is µ-preopen, so
U ⊂ iµ (cµ (A)) and A ⊂ cµ (U ) ,
so A ⊂ cµ (iµ (cµ (A))), that is, A is µ-β-open.
The following example shows that the converse of Proposition 3.9 need
not be true even in µ-spaces.
Example 3.10. Let
B = {{1, 2, 3} , {3, 4, 5} , {5, 6, 7}}
and let
µ = {∅, all possible unions of members of B} .
Then (X, µ) is a µ-space, where X = {1, 2, 3, 4, 5, 6, 7}. If A = {1, 2, 4}, then
A is µ-β-open because cµ (A) = {1, 2, 3, 4}, and thus
iµ (cµ (A)) = {1, 2, 3} , cµ (iµ (cµ (A))) = {1, 2, 3, 4} .
We will show now that there is no µ-preopen set U such that U ⊂ A ⊂
cµ (U ). We discuss the following cases:
(i) If U = ∅ then U is µ-preopen, but cµ (U ) = ∅.
(ii) If U = {1, 2} then U is not µ-preopen because cµ (U ) = {1, 2} and
iµ (cµ (U )) = ∅.
(iii) If U = {1, 4} then U is not µ-preopen because cµ (U ) = {1, 2, 3, 4} and
iµ (cµ (U )) = {1, 2, 3}.
(iv) If U = {2, 4} then U is not µ-preopen because cµ (U ) = {1, 2, 3, 4} and
iµ (cµ (U )) = {1, 2, 3}.
(v) If U = {1, 2, 4} then U is not µ-preopen because iµ (cµ (U )) = {1, 2, 3}.
Hence, there is no µ-preopen set U such that U ⊂ A ⊂ cµ (U ).
Proposition 3.11. Let A be a subset of a space (X, µ). If A is µ-preopen
then A is the intersection of a µ-regular open set and a µ-dense set.
Proof. Suppose that A is µ-preopen, that is, A ⊂ iµ (cµ (A)). Then
A = iµ (cµ (A)) ∩ (A ∪ (X\cµ (A))) .
Let
B = iµ (cµ (A)) and C = A ∪ (X\cµ (A)) .
Then B is µ-regular open by Proposition 1.5(vi), also
cµ (A) ⊂ cµ (C) since A ⊂ C
and
X\cµ (A) ⊂ C ⊂ cµ (C) .
Thus cµ (C) = X, i.e. C is µ-dense. Thus, A is the intersection of a µ-regular
open set and a µ-dense set.
Some properties of generalized open sets in generalized topological spaces
423
Corollary 3.12. Let A be a subset of a space (X, µ). If A is µ-preclosed,
then A is the union of a µ-regular closed set and a set with empty µ-interior.
The following example shows that the converse of Proposition 3.11 need
not be true even in µ-spaces.
Example 3.13. Consider the space of Example 3.10, and let
B = {1, 2, 3}
and C = {1, 2, 4, 5} .
Then B is µ-regular open as
cµ (B) = {1, 2, 3, 4} and iµ (cµ (B)) = {1, 2, 3} .
Also it is clear that C is µ-dense. However, B ∩ C = {1, 2} is not µ-preopen
as shown in Example 3.10.
Proposition 3.14. Let A be a subset of a space (X, µ). If A is µ-semiopen, then A is the intersection of a µ-regular closed set B and a set C such
that iµ (C) is µ-dense.
Proof. Suppose that A is µ-semi-open, that is, A ⊂ cµ (iµ (A)). Then
A = cµ (iµ (A)) ∩ (A ∪ (X\cµ (iµ (A)))) .
Let
B = cµ (iµ (A)) and C = A ∪ (X\cµ (iµ (A))) .
Then B is µ-regular closed by Proposition 1.5(vii), also
cµ (iµ (A)) ⊂ cµ (iµ (C)) since A ⊂ C
but
X\cµ (iµ (A)) ⊂ C
and X\cµ (iµ (A)) is µ-open, so
X\cµ (iµ (A)) ⊂ iµ (C) ⊂ cµ (iµ (C)) .
Thus cµ (iµ (C)) = X, i.e. iµ (C) is µ-dense. Hence, A is the intersection of
a µ-regular closed set and a set whose µ-interior is µ-dense.
Corollary 3.15. Let A be a subset of a space (X, µ). If A is µ-semiclosed, then A is the union of a µ-regular open set and a set whose µ-closure
has empty µ-interior.
The following example shows that the converse of Proposition 3.14 need
not be true even in µ-spaces.
Example 3.16. Consider the space of Example 3.10, and let
B = {4, 5, 6, 7}
and C = {3, 4, 5} .
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M. S. Sarsak
Then B is µ-regular closed, since {1, 2, 3} is µ-regular open as shown in
Example 3.13. Also, it is clear that iµ (C) = C is µ-dense. However, B ∩ C =
{4, 5} is not µ-semi-open as
iµ ({4, 5}) = ∅ and cµ (∅) = ∅.
Proposition 3.17. Let A be a subset of a space (X, µ). If A is µ-β-open,
then A is the intersection of a µ-regular closed set B and a µ-dense set C.
Proof. Suppose that A is µ-β-open, that is, A ⊂ cµ (iµ (cµ (A))). Then
A = cµ (iµ (cµ (A))) ∩ (A ∪ (X\cµ (A))) .
Let
B = cµ (iµ (cµ (A))) and C = A ∪ (X\cµ (A)) .
Then B is µ-regular closed by Proposition 1.5(vii), also
cµ (A) ⊂ cµ (C) since A ⊂ C
but
X\cµ (A) ⊂ C ⊂ cµ (C) .
Thus cµ (C) = X, i.e. C is µ-dense. Hence, A is the intersection of a
µ-regular closed set and a µ-dense set.
Corollary 3.18. Let A be a subset of a space (X, µ). If A is µ-β-closed,
then A is the union of a µ-regular open set and a set with empty µ-interior.
In [2], it is shown that a subset A of a topological space X is β-open if and
only if A = B ∩ C, where B is semi-open and C is dense. However, this is
not true for generalized topological spaces since the converse of Proposition
3.17 need not be true even in µ-spaces as following example shows.
Example 3.19. Consider the space of Example 3.10, and let
B = {4, 5, 6, 7}
and C = {3, 7} .
Then B is µ-regular closed as shown in Example 3.16. Also, it is clear that
C is µ-dense. However, B ∩ C = {7} is not µ-β-open as
cµ ({7}) = {6, 7} , iµ ({6, 7}) = ∅ and cµ (∅) = ∅.
4. Applications
Recall that a topological space X is called locally indiscrete if every open
subset of X is closed. The following easy result characterizes generalized
topological spaces (X, µ) in which every µ-open set is µ-closed in terms of
locally indiscrete topological spaces.
Proposition 4.1. A space (X, µ) is a locally indiscrete topological space
if and only if every µ-open subset of X is µ-closed.
Some properties of generalized open sets in generalized topological spaces
425
Proof. The necessity is clear. Assume that (X, µ) is a space in which every
µ-open set is µ-closed. It suffices to show that (X, µ) is a topological space.
(i) Since ∅ is µ-open, then by assumption, ∅ is µ-closed, that is, X is
µ-open.
(ii) Let U, V ∈ µ. Then by assumption, U, V are µ-closed, that is
X\U, X\V are µ-open, and thus
(X\U ) ∪ (X\V ) = X\ (U ∩ V )
is µ-open, and thus by assumption again, X\ (U ∩ V ) is µ-closed, that
is, U ∩ V is µ-open. Hence, (X, µ) is a topological space.
The proof of the following lemma is straightforward and thus omitted.
Lemma 4.2. Let A be a subset of a space (X, µ). If A is both µ-closed and
µ-preopen, then A is µ-open.
Since for a space (X, µ), α ⊂ σ, it might be natural to characterize spaces
in which α = σ. The following theorem provides several characterizations of
spaces (X, µ) in which α = σ.
Theorem 4.3. For a space (X, µ), the following are equivalent:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
every
every
every
every
every
every
every
every
µ-semi-open subset of X is µ-α-open, i.e. α = σ,
µ-semi-open subset of X is µ-preopen,
µ-β-open subset of X is µ-preopen, i.e. β = π,
µ-b-open subset of X is µ-preopen, i.e. b = π,
regular µ-semi-open subset of X is µ-preopen,
regular µ-semi-open subset of X is µ-regular open, i.e. r = rσ,
µ-regular closed subset of X is µ-preopen,
µ-regular closed subset of X is µ-open.
Proof. (i)→(ii) Clear since α ⊂ π.
(ii)→(iii): Let A be a µ-β-open subset of X. Then A ⊂ cµ (iµ (cµ (A))).
It follows from Proposition 1.5(vii) that B = cµ (iµ (cµ (A))) is µ-regular
closed and thus µ-semi-open. By (ii), B is µ-preopen. Thus
A ⊂ B ⊂ iµ (cµ (B)) = iµ (B) .
Also it is clear that
B ⊂ cµ (A) and thus iµ (B) ⊂ iµ (cµ (A)) .
Therefore
A ⊂ iµ (cµ (A)) .
Hence, A is µ-preopen.
(iii)→(iv): Follows since b ⊂ β.
(iv)→(v): It follows from Proposition 2.12 that rσ ⊂ σ, but σ ⊂ b, thus
rσ ⊂ b. Therefore, the result follows from (iii).
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M. S. Sarsak
(v)→(vi): Since every regular µ-semi-open set is µ-semi-closed (Proposition 2.12), it follows from (iv) that a regular µ-semi-open set is both µ-semiclosed and µ-preopen. Thus by Proposition 1.6, (v) follows.
(vi)→(vii): Follows from Proposition 1.6 and Remark 2.11.
(vii)→(viii): Follows from Lemma 4.2.
(viii)→(i): Let A be a µ-semi-open subset of X. Then by Corollary
2.9, cµ (A) is µ-regular closed. Thus by (vii), cµ (A) is µ-open, that is,
cµ (A) ⊂ iµ (cµ (A)), i.e. A is µ-preopen. Since A ∈ σ ∩ π = α, (i) follows.
Corollary 4.4. For a space (X, µ), the following are equivalent:
(i) α = σ,
(ii) every regular µ-semi-open subset of X is µ-preclosed,
(iii) every regular µ-semi-open subset of X is µ-regular closed.
Proof. Follows from Remark 2.13 and Theorem 4.3.
Corollary 4.5. For a space (X, µ), the following are equivalent:
(i) α = σ,
(ii) every regular µ-semi-open subset of X is µ-preclopen, i.e. µ-preopen
and µ-preclosed,
(iii) every regular µ-semi-open subset of X is µ-clopen, i.e. µ-open and
µ-closed.
Proof. Follows from Theorem 4.3 and Corollary 4.4.
Proposition 4.6. For a space (X, µ), the following are equivalent:
(i) every µ-preopen subset of X is µ-α-open, i.e. α = π,
(ii) every µ-preopen subset of X is µ-semi-open.
Proof. Follows from Proposition 1.2 (iii).
Proposition 4.7. For a space (X, µ), consider the following properties:
(i) every µ-β-open subset of X is µ-semi-open, i.e. β = σ,
(ii) every µ-b-open subset of X is µ-semi-open, i.e. b = σ,
(iii) every µ-preopen subset of X is µ-semi-open.
Then (i)→(ii), (ii)→(iii), and thus (i)→(iii).
Proof. Follows from Remark 2.2.
Question. Are the reverse implications of Proposition 4.7 true?
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DEPARTMENT OF MATHEMATICS
FACULTY OF SCIENCE
THE HASHEMITE UNIVERSITY
P.O. BOX 150459, ZARQA 13115, JORDAN
E-mail: [email protected]
Received April 28, 2011.