1. Introduction
In 1937, regular open sets were introduced and used to define the semi-regularization space of a topological space. Throughout this paper,
and
stand for topological spaces with no separation axioms assumed unless otherwise stated. For a subset A of X, the closure of A and the interior of A will be denoted by
and
respectively. Stone [1] defined a subset A of a space X to be a regular open if
. Norman Levine [2] defined a subset A of a space X to be a semi-open if
, or equivalently, a set A of a space X will be termed semiopen if and only if there exists an open set
such that
. Mashhour et al. [3] defined a subset A of a space X to be a preopen if
. Njastad [4] defined a subset A of a space X to be an
-open if
. The complement of a semi-open (resp., regular open) set is said to be semi-closed [5] (resp., regular closed). The intersection of all semi-closed sets of X containing A is called the semi-closure [6] of A. The union of semi-open sets of X contained in A is called the semi-interior of A. Joseph and Kwack [7] introduced the concept of
-semi open sets using semi-open sets to improve the notion of
-closed spaces. Also Joseph and Kwack [7] introduced that a subset A of a space X is called
-semi-open if for each
, there exists a semi-open set
such that
. It is well-known that, a space X is called
if to each pair of distinct points x, y of X, there exists a pair of open sets, one containing x but not y and the other containing y but not x, as well as is
if and only if for any point
, the singleton set
is closed. A space X is regular if for each
and each open set G containing x, there exists an open set H such that
. Ahmed [8] defined a topological space
to be s**-normal if and only if for every semi-closed set F and every semi-open set G containing F, there exists an open set H such that
. In 1968, Velicko [9], defined the concepts of
-open and
-open as, a subset A of a space X is called
-open (resp.,
-open) if for each
, there exists an open set
such that
(resp.,
). Di Maio and Noiri [10] introduced that a subset A of a space X is called semi-
-open if for each
, there exists a semi-open set G such that
. The family of all open (resp., semi-open,
-open, preopen, θ-semi-open, semi-θ-open, θ-open, δ-open, regular open, semi-closed and regular closed) subsets of a topological space
are denoted by
(resp.,
,
,
,
,
,
,
,
,
and
).
Definition 1.1. [11] A subset A of a space X is called b-open if
. The family of all b-open subsets of a topological space
is denoted by
or (Briefly.
).
In 1999, J. Dontchev and T. Noiri [12] have shown the following lemma:
Lemma 1.2. For a subset A of a space
, the following conditions are equivalent:
1) ![](https://www.scirp.org/html/7-5300282\d1a7b40c-62ad-4c29-bb49-68e9ffff05a6.jpg)
2) ![](https://www.scirp.org/html/7-5300282\423bdb7f-008d-46a2-bbfd-6d16e4084242.jpg)
3) ![](https://www.scirp.org/html/7-5300282\17b8a9c5-fedd-4d9a-ab67-41a814cc1952.jpg)
4) ![](https://www.scirp.org/html/7-5300282\0c300ecc-2cea-46aa-9641-3e7cd25fca46.jpg)
Theorem 1.3. [13] If
is s**-normal, then
.
We recall that a topological space X is said to be extremally disconnected [14] if
is open for every open set G of X.
Definition 1.4. [15] A space X is called locally indiscrete if every open subset of X is closed.
Theorem 1.5. [13] A space X is extremally disconnected if and only if
.
Theorem 1.6. [15] A space
is extremally disconnected if and only if
.
2. Bc-Open Sets
In this section, we introduce a new class of b-open sets called Bc-open sets in topological spaces.
Definition 2.1. A subset A of a space X is called Bcopen if for each
, there exists a closed set F such that
. The family of all Bc-open subsets of a topological space
is denoted by
or (Briefly.
).
Proposition 2.2. A subset A of a space X is Bc-open if and only if A is b-open and it is a union of closed sets. That is
where A is b-open set and
is closed sets for each
.
Proof. Obvious.
It is clear from the definition that every Bc-open subset of a space X is b-open, but the converse is not true in general as shown by the following example.
Example 2.3. Consider
with the topology
. Then the family of closed sets are:
. We can find easily the following families:
![](https://www.scirp.org/html/7-5300282\3163a1a8-f15f-4961-9928-480ad2eefa02.jpg)
and
.
Then
but ![](https://www.scirp.org/html/7-5300282\394361a0-6873-457e-9c3b-6b2f48d547db.jpg)
The next example notices that a Bc-open set need not be a closed set.
Example 2.4. Consider the space R with usual topology, if
such that
, then
is Bc-open set, but it is not closed.
The following result shows that the arbitrary union of Bc-open sets in a topological space
is Bc-open.
Proposition 2.5. Let
be a collection of Bc-open sets in a topological space X. Then
is Bc-open.
Proof. Let
be a Bc-open set for each
, then
is
-open and hence
is b-open. Let
, there exist
such that
. Since
is b-open for each
, there exists a closed set
such that
![](https://www.scirp.org/html/7-5300282\2c7e6b86-0939-4e51-9d4c-a4d58f4e593f.jpg)
so
Therefore,
is Bc-open set.
The following example shows that the intersection of two Bc-open sets need not be Bc-open set.
Example 2.6. Consider the space
as in example 2.3, There
and
, but ![](https://www.scirp.org/html/7-5300282\e0a1e7db-fc43-44eb-bfca-f8dbc7d36d19.jpg)
From the above example we notice that the family of all Bc-open subsets of a space X is a supratopology and need not be a topology in general.
The following result shows that the family of all Bcopen sets will be a topology on X.
Proposition 2.7. If the family of all b-open sets of a space X is a topology on X, then the family of Bc-open is also a topology on X.
Proof. Clearly
and by Proposition 2.5 the union of any family of Bc-open sets is Bc-open. To complete the proof it is enough to show that the finite intersection of Bc-open sets is Bc-open set. Let A and B be two Bc-open sets then A and B are
-open sets. Since
is a topology on X, so
is b-open. Let
, then
and
, so there exists F and E such that
and
this implies that
. Since any intersection of closed sets is closed,
is closed set. Thus
is Bc-open set. This completes the proof.
Proposition 2.8. The set A is Bc-open in the space
if and only if for each
, there exists a Bcopen set B such that
.
Proof. Assume that A is Bc-open set in the
, then for each
, put
is Bc-open set containing x such that
.
Conversely, suppose that for each
, there exists a Bc-open set B such that
, thus
where
for each x, therefore A is Bc-open set.
In the following proposition, the family of b-open sets is identical to the family of Bc-open sets.
Proposition 2.9. If a space X is
-space, then the families ![](https://www.scirp.org/html/7-5300282\c4d6a234-ca8b-4cd6-a204-16be0326d59d.jpg)
Proof. Let A be any subset of a space X and
, if
, then
, then for each
. Since a space X is
, then every singleton is closed set and hence
. Therefore
. Hence
, but
generally, therefore
.
Proposition 2.10. Every
-semi open set of a space X is Bc-open set.
Proof. Let A be a
-semi open set in X, then for each
, there exists a semi-open set G such that
, so
for each
implies that
which is semi-open set and
is a union of closed sets, by Proposition 2.2, A is Bc-open set.
The following example shows that the converse of the above Proposition may not be true in general.
Example 2.11. Since a space X with cofinite topology is T1, and then the family of b-open and Bc-open sets are identical. Hence any open set G is Bc-open but not
- semi open.
The proof of the following corollaries is clear from their definitions.
Corollary 2.12. Every
-open set is Bc-open.
Corollary 2.13. Every regular-closed is Bc-open set.
Proposition 2.14. If a topological space
is locally indiscrete, then
.
Proof. Let A be any subset of a space X and
, if
, then
. If
, then
. Since X is locally indiscrete, then
is closed and hence
, this implies that for each
,
. Therefore, A is Bc-open set. Hence
.
Remark 2.15. Since every open set is semi-open, it follows that if a topological space
is
or locally indiscrete, then ![](https://www.scirp.org/html/7-5300282\5b5e624b-292b-46be-808a-8ae5b8e02dab.jpg)
Proposition 2.16. Let
be a topological space, if X is regular, then ![](https://www.scirp.org/html/7-5300282\1495a3c7-1bb4-45fd-a32d-438ecbbf55bd.jpg)
Proof. Let A be any subset of a space X, and A is open, if
, then
. If
, since X is regular, so for each
, there exists an open set G such that
. Thus we have
. Since
and hence
, therefore
.
Proposition 2.17. Let
be an extremally disconnected space. If
, then
.
Proof. Let
. If
, then
. If
. Since a space X is extremally disconnected, then by Theorem 1.5,
. Hence
. But
in general. Therefore,
.
Corollary 2.18. Let
be an extremally disconnected space. If
, then
.
Proof. The proof is directly from Proposition 2.28 and the fact that ![](https://www.scirp.org/html/7-5300282\52d5ca8e-12ea-47e2-beec-7ce0954258ab.jpg)
Proposition 2.19. Let
be an s**-normal space. If
, then
.
Proof. Let
. If
, then
. If
, since a space X is s**-normal, then by Theorem 1.3,
. Hence
. But
in general. Therefore,
.
Proposition 2.20. For any subset A of a space
and
. The following conditions are equivalent:
1) A is regular closed.
2) A is closed and Bc-open.
3) A is closed and b-open.
4) A is α-closed and b-open.
5) A is pre-closed and b-open.
Proof. Follows from Lemma 1.2.
Definition 2.21. A subset B of a space X is called Bcclosed if
is Bc-open. The family of all Bc-closed subsets of a topological space
is denoted by
or (Briefly,
).
Proposition 2.22. A subset B of a space X is Bc-closed if and only if B is a b-closed set and it is an intersection of open sets.
Proof. Clear.
Proposition 2.23. Let
be a collection of Bc-closed sets in a topological space X. Then
is Bc-closed.
Proof. Follows from Proposition 2.5.
The union of two Bc-closed sets need not be Bc-closed as is shown by the following counterexample.
Example 2.24. In Example 2.3, the family of Bcclosed subset of X is:
. Here
and
, but
.
All of the following results are true by using complement.
Proposition 2.25. If a space X is
, then
![](https://www.scirp.org/html/7-5300282\31dbdacc-bc10-4555-9fe4-a2eb3c058393.jpg)
Proposition 2.26. For any subset B of a space X. If
, then
.
Corollary 2.27. Each
-closed set is Bc-closed.
Corollary 2.28. Each regular open set is Bc-closed.
Proposition 2.29. If a topological space
is locally indiscrete, then
.
Proposition 2.30. Let
be a topological space, if X is regular or locally indiscrete, then the family of closed sets is a subset of the family of Bc-closed sets.
Proposition 2.31. Let
be any extremally disconnected space. If
, then
.
Corollary 2.32. Let
be an extremally disconnected space. If
, then
.
Proposition 2.33. Let
be a s**-normal space. If
, then
.
Proposition 2.34. For any subset B of a space
and
. The following conditions are equivalent:
1) B is regular open.
2) B is open and Bc-closed.
3) B is open and b-closed.
4) B is α-open and b-closed.
5) B is preopen and b-closed.
Diagram 1 shows the relations among
,
,
,
,
,
,
,
and
.
![](https://www.scirp.org/html/7-5300282\e2e48860-208d-446a-b9d3-ba9c860446a9.jpg)
Diagram 1.
3. Some Properties of Bc-Open Sets
In this section, we define and study topological properties of Bc-neighborhood, Bc-interior, Bc-closure and Bcderived of a set using the concept of Bc-open sets.
Definition 3.1. Let
be a topological space and
, then a subset N of X is said to be Bc-neighborhood of
, if there exists a
-open set U in X such that
.
Proposition 3.2. In a topological space
, a subset A of X is Bc-open if and only if it is a Bcneighbourhood of each of its points.
Proof. Let
be a Bc-open set, since for every
and A is Bc-open. this shows A is a Bc-neighborhood of each of its points.
Conversely, suppose that A is a Bc-neighborhood of each of its points. Then for each
, there exists
such that
. Then
. Since each Bx is Bc-open. It follows that A is Bc-open set.
Proposition 3.3. For any two subsets A, B of a topological space
and
, if A is a Bc-neighborhood of a point
, then B is also Bc-neighborhood of the same point
.
Proof. let A be a Bc-neighborhood of
, and
, then by Definition 2.1, there exists a Bc-open set U such that
, this implies that B is also a Bc-neighborhood of x.
Remark 3.4. Every Bc-neighborhood of a point is bneighborhood, it follows from every Bc-open set is bopen.
Definition 3.5. Let A be a subset of a topological space
, a point
is said to be Bc-interior point of
, if there exist a Bc-open set
such that
. The set of all Bc-interior points of A is called Bc-interior of A and is denoted by
.
Some properties of the Bc-interior of a set are investigated in the following theorem.
Theorem 3.6. For subsets A, B of a space X, the following statements hold.
1)
is the union of all Bc-open sets which are contained in A.
2)
is
-open set in X.
3)
is
-open if and only if
.
4)
.
5)
and
.
6)
.
7) If
, then
.
8) If
, then
.
9)![](https://www.scirp.org/html/7-5300282\70d5833f-ec8b-4d5f-8238-d151ab9e6285.jpg)
10)
.
Proof. 7) Let
and
, then by Definition 3.5, there exists a Bc-open set
such that
implies that
. thus
.
The other parts of the theorem can be proved easily.
Proposition 3.7. For a subset A of a topological space
, then
.
Proof. This follows immediately since all
-open set is b-open.
Definition 3.8. Let A be a subset of a space X. A point
is said to be Bc-limit point of A if for each Bcopen set U containing
. The set of all Bc-limit points of A is called a Bc-derived set of A and is denoted by
.
Proposition 3.9. Let A be a subset of X, if for each closed set F of X containing x such that
, then a point
is Bc-limit point of A.
Proof. Let U be any Bc-open set containing x, then for each
, there exists a closed set F such that
. By hypothesis, we have
. Hence
. Therefore, a point
is Bc-limit point of A.
Some properties of Bc-derived set are stated in the following theorem.
Theorem 3.10. Let A and B be subsets of a space X. Then we have the following properties:
1)
.
2) If
, then
.
3) If
, then
.
4)
.
5)
.
6)
.
7)
.
Proof. We only prove 6), 7), and the other parts can be proved obviously.
6) If
and
is a Bc-open set containing x, then
. Let
. Then, since
and
. Let
. Then,
for
and
. Hence,
. Therefore, ![](https://www.scirp.org/html/7-5300282\1b72f1a4-61b3-4550-929f-64f03b65cc4c.jpg)
7) Let
. If
, the result is obvious. So, let
, then, for Bcopen set
containing
. Thus,
or
. Now, it follows similarly from 1) that
. Hence,
. Therefore, in any case,
.
Corollary 3.11. For a subset A of a space X, then
.
Proof. It is sufficient to recall that every Bc-open set is b-open.
Definition 3.12. For any subset A in the space X, the Bc-closure of A, denoted by
, is defined by the intersection of all Bc-closed sets containing A.
Proposition 3.13. A subset A of a topological space X is Bc-closed if and only if it contains the set of its Bclimit points.
Proof. Assume that A is Bc-closed and if possible that x is a Bc-limit point of A which belongs to
, then
is Bc-open set containing the Bc-limit point of A, therefore
, which is a contradiction.
Conversely, assume that A contains the set of its Bclimit points. For each
, there exists a Bc-open set U containing x such that
, that is
by Proposition 2.8,
is Bc-open set and hence A is Bc-closed set.
Proposition 3.14. Let A be a subset of a space X, then
.
Proof. Since
, then
.
On the other hand. To show that
, since
is the smallest Bc-closed set containing A, so it is enough to prove that
is Bc-closed. Let
. This implies that
and
. Since
, there exists a Bc-open set
of
which contains no point of A other than x but
. So Gx contains no point of A, which implies
. Again, Gx is a Bc-open set of each of its points. But as Gx does not contain any point of A, nopoint of Gx can be a Bc-limit point of A. Therefore, nopoint of Gx can belong to
. This implies that
. Hence, it follows that
![](https://www.scirp.org/html/7-5300282\f42e2eba-f8b9-45cc-94c0-e7b811ff1099.jpg)
Therefore,
is Bc-closed. Hence
. Thus
.
Corollary 3.15. Let A be a set in a space X. A point
is in the Bc-closure of A if and only if
for every Bc-open set U containing x.
Proof. Let
. Then
, where F is Bc-closed with
. So
and
is a Bc-open set containing x and hence
![](https://www.scirp.org/html/7-5300282\d25ab30c-e0a3-4d3f-95c0-7bc2d6f6451c.jpg)
Conversely, suppose that there exists a Bc-open set containing x with
. Then
and
is a Bc-closed. Hence
.
Proposition 3.16. Let A be any subset of a space X. If
for every closed set F of X containing x, then the point x is in the Bc-closure of A.
Proof. Suppose that U be any Bc-open set containing x, then by Definition 2.1, there exists a closed set F such that
. So by hypothesis
implies
for every Bc-open set U containing x. Therefore
.
Here we introduce some properties of Bc-closure of the sets.
Theorem 3.17. For subsets A, B of a space X, the following statements are true.
1) The Bc-closure of A is the intersection of all Bcclosed sets containing A.
2)![](https://www.scirp.org/html/7-5300282\45222d6d-9fe4-42f3-969f-592ef7b20ad0.jpg)
3)
-closed set in X 4)
is Bc-closed set if and only if ![](https://www.scirp.org/html/7-5300282\71204f49-0e7b-4627-b4db-9eb975e7d608.jpg)
5)![](https://www.scirp.org/html/7-5300282\14c38402-4c11-4b70-9837-bf4465157f84.jpg)
6)
and
.
7) If
, then ![](https://www.scirp.org/html/7-5300282\0f04cde8-794d-4aa1-8216-4801f7c82719.jpg)
8) If ![](https://www.scirp.org/html/7-5300282\67ee603a-32c3-4fb8-bf79-6447e2651aa3.jpg)
9)![](https://www.scirp.org/html/7-5300282\f5bc7c33-0e2f-4111-8e8a-9a9063cf619f.jpg)
10)
.
Proof. Obvious.
Proposition 3.18. For any subset A of a topological space X. The following statements are true.
1)![](https://www.scirp.org/html/7-5300282\b67ffc26-2c56-49af-b56c-7210befacc7d.jpg)
2)![](https://www.scirp.org/html/7-5300282\dc5269cf-3207-418d-9082-a8ab5cb29e2f.jpg)
3)![](https://www.scirp.org/html/7-5300282\58249caf-d069-4564-a379-2984b791290c.jpg)
4)![](https://www.scirp.org/html/7-5300282\bd676f90-4a5a-4703-b7e9-f0b4ea316703.jpg)
Proof. We only prove 1), the other parts can be proved similarly. For any point
,
implies that
, then for each
containing
, then
. Thus
.
Conversely, by reverse the above steps, we can prove this part.
Remark 3.19. If
is a subset of a topological space X. Then
![](https://www.scirp.org/html/7-5300282\6707f48a-a636-4384-b427-e3e01a89abcf.jpg)
Proof. Obvious.
4. Bc-Compactness
In this section, we introduce and investigate new class of space named Bc-compact.
Definition 4.1. A filter base
in a topological space
Bc-converges to a point
if for every Bcopen set V containing x, there exists an
such that
.
Definition 4.2. A filter base
in a topological space
Bc-accumulates to a point
if
, for every
-open set V containing
and every
.
Proposition 4.3. Let
be a filter base in a topological space
. If
Bc-converges to a point
, then
rc-converges to a point x.
Proof. Suppose that
Bc-converges to a point
. Let V be any regular closed set containing x, then
. Since
Bc-converges to a point
, there exists an
such that
. This shows that
rc-converges to a point x.
In general the converse of the above proposition is not necessarily true, as the following example shows.
Example 4.4. Consider the space
. Let
. Then
rc-converges to 0, but
does not Bc-converges to 0, because the set
is Bc-open containing 0, there exist no
such that
.
Corollary 4.5. Let
be a filter base in a topological space
. If
Bc-accumulates to a point
, then
rc-accumulates to a point x.
Proof. Similar to Proposition 4.3.
Proposition 4.6. Let
be a filter base in a topological space
and E is any closed set containing
. If there exists an
such that
, then
Bcconverges to a point
.
Proof. Let
be any Bc-open set containing
, then for each
, there exists a closed set E such that
. By hypothesis, there exists an
such that
which implies that
. Hence
Bc-converges to a point
.
Proposition 4.7. Let
be a filter base in a topological space
and E is any closed set containing
, such that
for each
, then
is Bcaccumulation to a point
.
Proof. The proof is similar to Proposition 4.6.
Definition 4.8. We say that a topological space
is Bc-compact if for every Bc-open cover
of X, there exists a finite subset
of
such that
.
Theorem 4.9. If every closed cover of a space X has a finite subcover, then X is Bc-compact.
Proof. Let
be any Bc-open cover of X, and
, then for each
, there exists a closed set
such that
. So the family
is a cover of X by closed set, then by hypothesis, this family has a finite subcover such that
.
Therefore,
. Hence X is Bccompact.
Proposition 4.10. If a topological space
is bcompact, then it is Bc-compact.
Proof. Let
be any Bc-open cover of X. Then
is b-open cover of X. Since X is bcompact, there exists a finite subset
of
such that
. Hence X is Bc-compact.
Proposition 4.11. Every Bc-compact
-space is bcompact.
Proof. Suppose that X is
and Bc-compact space. Let
be any b-open cover of X. Then for every
, there exists
such that
. Since X is
, by Since X is Bc-compact, so there exists a finite subset
of
in X such that
. Hence X is b-compact.
The next corollary is an immediate consequence of Proposition 4.10 and 4.11.
Corollary 4.12. Let X be a
-space. Then X is Bccompact if and only if X is b-compact.
Proposition 4.13. Let a topological space
be locally indiscrete. If X is Bc-compact then X is s-compact.
Proof. Follows from Proposition 2.14.
Proposition 4.14. If a topological space
is Bccompact, then it is rc-compact.
Proof. Let
be any regular closed cover of X. Then
is a Bc-open cover of X. Since X is
-compact, there exists a finite subset
of
such that
. Hence X is rc-compact.
Proposition 4.15. Let a topological space
be regular. If X is Bc-compact, then it is compact.
Proof. Let
be any open cover of X. By Proposition 2.16,
forms a Bc-open cover of X. Since X is Bc-compact, there exists a finite subset
of
such that
. Hence X is compact.
Proposition 4.16. Let X be an almost regular space. If X is Bc-compact, then it is nearly compact.
Proof. Let
be any regular open cover of X. Since X is almost regular space, then, for each
and regular open
there exists an open set Gx such that
But
is regulaclosed for each
, this implies that the family
is Bc-open cover of X, since X is Bc - compact, then there exists a subfamily
such that
. Thus X is nearly compact.