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UDK 515.122

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ON b-OPEN SETS

Dimitrije Andrijević

Abstract. A new class of generalized open sets in a topological space, called b-open sets, is introduced and studied. This class is contained in the class of semi-preopen sets and contains all semi-open sets and all preopen sets. It is proved that the class of b-open sets generates the same topology as the class of preopen sets.

1. Introduction

In recent years a number of generalizations of open sets have been considered. Four of these notions were defined similarly using the closure operator cl and the interior operator int in the following way.

Definition 1. A subset S of a space X is called

(1) an α-set if S ⊂ int(cl(int S)),

(2) semi-open if S ⊂ cl(int S),

(3) preopen if S ⊂ int(cl S),

(4) semi-preopen if S ⊂ cl(int(cl S)).

The first three notions are due to Njåstad [10], Levine [8] and Mashhour et al. [9], respectively. The concept of a preopen set was introduced by Corson and Michael [6] who used the term “locally dense”. The fourth concept was introduced by Abd El-Monsef et al. [1] under the name “β-open”, and in [3] these sets were called semi-preopen sets. We denote the classes of these sets in a space (X, T ) by T_{α}, SO(X), P O(X) and SP O(X) respectively. All of them are larger than T and closed under forming arbitrary unions. Njåstad [10] showed that T_{α} is a topology on X. In general, SO(X) need not be a topology on X, but the intersection of a semi-open set and an open set is semi-open. The same holds for P O(X) and SP O(X). The complement of a semi-open set is called semi-closed. Thus S is semi-closed if and only if int(cl S) ⊂ S. Preclosed and semi-preclosed sets are similarly defined. For a subset S of a space X the semi-closure (resp. preclosure,

AMS Subject Classification: 54 A 10

Keywords and phrases: topological space, generalized open sets, α-set, b-open, semi-open, preopen, semi-preopen set

60 D. Andrijević

semi-preclosure) of S, denoted by scl S (resp. pcl S, spcl S) is the intersection of all semi-closed (resp. preclosed, semi-preclosed) subsets of X containing S. Dually, the semi-interior (resp. preinterior, semi-preinterior) of S, denoted by sint S, (resp. pint S, spint S), is the union of all semi-open (resp. preopen, semi-preopen) subsets of X contained in S. By cl_{α} and int_{α} we denote the closure and the interior operator in (X, T_{α}).

An extensive study of these operators was done in [3]. We recollect some of the relations that, together with their duals, we shall use in the sequel.

Proposition 1.1. Let S be a subset of a space X. Then:

(1) cl_{α}S = S ∪ cl(int(cl S)), int_{α}S = S ∩ int(cl(int S)),

(2) scl S = S ∪ int(cl S), sint S = S ∩ cl(int S),

(3) pcl S = S ∪ cl(int S), pint S = S ∩ int(cl S),

(4) spcl S = S ∪ int(cl(int S)), spint S = S ∩ cl(int(cl S)).

Proposition 1.2. Let S be a subset of a space X. Then:

(1) scl(sint S) = sint S ∪ int(cl(int S)).

(2) pcl(pint S) = pint S ∪ cl(int S).

(3) spcl(spint S) = spint(spcl S).

(4) int(scl S) = pint(cl S) = pint(scl S) = scl(pint S) = int(cl S).

(5) int(pcl S) = scl(int S) = spcl(int S) = int(spcl S) = int(cl(int S)).

2. Some properties of b-open sets.

Now we consider a new class of generalized open sets. Definition 2. A subset S of a space X is called b-open if S ⊂ cl(int S) ∪

int(cl S). The class of all b-open sets in X will be denoted by B(X).

It is obvious that P O(X) ∪ SO(X) ⊂ BO(X) ⊂ SP O(X) and we shall show that the inclusions cannot be replaced with equalities.

Example. Consider the set R of real numbers with the usual topology, and let S = [0, 1] ∪ ((1, 2) ∩ Q) where Q stands for the set of rational numbers. Then S is b-open but neither semi-open nor preopen. On the other hand, let T = [0, 1)∩Q. Then T is semi-preopen but not b-open.

Proposition 2.1. For a subset S of a space X the following are equivalent:

(a) S is b-open.

(b) S = pint S ∪ sint S.

(c) S ⊂ pcl(pint S). Proof. (a) ⇒ (b): Let S be b-open, that is S ⊂ cl(int S) ∪ int(cl S). Then

by Proposition 1.1 we have pint S ∪ sint S = (S ∩ int(cl S)) ∪ (S ∩ cl(int S)) = S ∩ (int(cl S) ∪ (cl(int S)) = S.

On b-open sets 61

(b) ⇒ (c): Propositions 1.1 and 1.2 imply S = pint S ∪ sint S = pint S ∪ (S ∩ cl(int S)) ⊂ pint S ∪ cl(int S) = pcl(pint S).

(c) ⇒ (a): By Propositions 1.2 and 1.1 we have S ⊂ pcl(pint S) = pint S ∪ cl(int S) ⊂ int(cl S) ∪ cl(int S), and so S is b-open.

Remark 1. It follows from (b) that every b-open set can be represented as a union of a preopen set and a semi-open set. Since pint S \ sint S = pint S \ (S ∩ cl(int S)) = pint S \ cl(int S) is preopen, we can always have a disjoint union.

Remark 2. It was shown in [3] that S is semi-preopen if and only if S ⊂ sint(scl S). It is easy to prove that the condition S ⊂ scl(sint S) characterizes the semi-open sets, and the condition S ⊂ pint(pcl S) characterizes the preopen sets.

The next result is obvious.

Proposition 2.2. Let S be a b-open set such that int S = ∅. Then S is preopen.

Proposition 2.3. (a) The union of any family of b-open sets is a b-open set.

(b) The intersection of an open and a b-open set is a b-open set.

Proof. The statements are proved by using the same method as in proving the corresponding results for the other three classes of generalized open sets. (See [3].)

It was shown in [2] that cl_{α}int_{α}S = cl(int S) and int_{α}cl_{α}S = int(cl S) for any S ⊂ X. We deduce from it the following result, analogous to those established for the other three classes of generalized open sets.

Proposition 2.4. Let (X, T ) be a space. Then:

(1) T and T_{α} have the same class of b-open sets.

(2) The intersection of an α-set and a b-open set is a b-open set.

Definition 3. A subset S of a space X is called b-closed if X \ S is b-open. Thus S is b-closed if and only if int(cl S) ∩ cl(int S) ⊂ S.

Definition 4. If S is a subset of a space X the b-closure of S, denoted by bcl S, is the smallest b-closed set containing S. The b-interior of S, denoted by bint S, is the largest b-open set contained in S.

Proposition 2.5. Let S be a subset of a space X. Then:

(1) bcl S = scl S ∩ pcl S.

(2) bint S = sint S ∪ pint S.

Proof. We shall prove only the first statement. Since bcl S is a b-closed set, we have bcl S ⊃ int(cl(bcl S)) ∩ cl(int(bcl S)) ⊃ int(cl S) ∩ cl(int S) and so bcl S ⊃ S ∪ (int(cl S) ∩ cl(int S)) = scl S ∩ pcl S by Proposition 1.1. The reverse inclusion is clear.

The last result as well as Propositions 1.1 and 1.2, enable us to relate the operators of b-closure and b-interior to the other operators defined by generalized open sets.

Proposition 2.6. Let S be a subset of a space X. Then:

(1) bcl(int S) = int(bcl S) = int(cl(int S)).

(2) bint(cl S) = cl(bint S) = cl(int(cl S)).

(3) bcl(sint S) = scl(sint S).

(4) bint(scl S) = sint(scl S).

(5) sint(bcl S) = scl S ∩ cl(int S).

(6) scl(bint S) = sint S ∪ int(cl S).

(7) pint(bcl S) = bcl(pint S) = pint(pcl S).

(8) pcl(bint S) = bint(pcl S) = pcl(pint S).

(9) spint(bcl S) = bcl(spint S) = sint(scl S) ∩ pcl S.

(10) spcl(bint S) = bint(spcl S) = scl(sint S) ∪ pint S.

We conclude this section with

Proposition 2.7. Let S be a subset of a space X. Then bint(bcl S) = bcl(bint S).

Proof. The statement follows from Propositions 1.2, 2.5 and 2.6.

3. On the topology generated by b-open sets.

Although none of SO(X), P O(X), SP O(X) and BO(X) is a topology on X, each of these classes generates a topology in a natural way. Let T (S) = { V ⊂ X | V ∩ S ∈ S whenever S ∈ S }, where S stands for SO(X), P O(X), SP O(X) and BO(X) respectively. Clearly T (S) is a topology on X larger than T . Njåstad [10] showed that T (S) = T_{α} for S = SO(X). The topology generated by P O(X) was studied in [4] and denoted by T_{γ}. The closure and the interior of a set S in (X, T_{γ}) are denoted by cl_{γ}S and int_{γ}S. It was proved in [7] that T (S) = T_{γ} for S = SP O(X). The topology generated by b-open sets will be denoted by T_{b} and we shall prove that T_{b} = T_{γ}. We first recollect some results.

Proposition 3.1. [7] The intersection of a semi-open and a preopen set is a semi-preopen set.

Proposition 3.2. [7] For a space (X, T ) and x ∈ X the following are equivalent:

(a) {x} ∈ SP O(X).
(b) {x} ∈ P O(X).
(c) {x} ∈ T_{γ}.

Proposition 3.3. [4] Let S be a subset of a space X. Then cl_{γ} int S = cl(int S).

Proposition 3.4. [5] Let S be a subset of a space X. Then S ∈ T_{γ} if and only if S = G ∪ H with G ∈ T_{α} and {h} ∈ P O(X) for every h ∈ H.

The last result has an immediate consequence.

Corollary 3.5. Let V ∈ T_{γ} and G ∈ SO(X). Then V ∩ G ∈ BO(X).

Now we are in a position to prove the first part of our main result.

Theorem 3.6. Let (X, T ) be a space. Then T_{γ} ⊂ T_{b}.

Proof. Let V ∈ T_{γ} and S ∈ BO(X). Then S = pint S ∪ sint S by Proposition 2.1. Hence V ∩ S = (V ∩ pint S) ∪ (V ∩ sint S) is b-open by Proposition 3.5, and so V ∈ T_{b}.

In order to prove the reverse inclusion we first establish two lemmas.

Lemma 3.7. Let (X, T ) be a space and V ∈ T_{b}. Then S = V \ int(cl(int V)) is a preopen set.

Proof. Since int(cl(int V)) is semi-closed, S is b-open. On the other hand, int S = ∅ and so S is preopen by Proposition 2.2.

Lemma 3.8. Let (X, T ) be a space and V ∈ T_{b}. Then sint V = int_{α}V.

Proof. Let x ∈ sint V = V ∩ cl(int V) and suppose x ∉ int(cl(int V)). Then {x} ∪ int V is semi-open and V \ int(cl(int V)) is preopen by Lemma 3.7. Hence {x} = ({x} ∪ int V) ∩ (V \ int(cl(int V))) is semi-preopen by Proposition 3.1, and so {x} ∈ T_{γ} by Proposition 3.2. On the other hand, {x} ∈ cl_{γ} int V by Proposition 3.3, a contradiction. Therefore x ∈ int(cl(int V)) that is x ∈ int_{α}V.

Theorem 3.9. Let (X, T ) be a space. Then T_{b} = T_{γ}.

Proof. Let V ∈ T_{b} and S ∈ P O(X). Put G = V ∩ cl(int V) and H = V \ cl(int V). Then by previous lemma V = int_{α}V ∪ H. Since H ∈ T_{b} and int H = ∅, we have that H ∩ S ∈ P O(X) by Proposition 2.2. Therefore V ∩ S = (int_{α}V ∩ S) ∪ (H ∩ S) is preopen and so V ∈ T_{γ}.

REFERENCES

[1] M.E. Abd El-Monsef, S.N. El-Deeb and R.A. Mahmoud, β-open sets and β-continuous mappings, Bull. Fac. Sci. Assiut Univ. 12 (1983), 77–90.

[2] D. Andrijević, Some properties of the topology of α-sets, Mat. Vesnik 36 (1984), 1–10.

[3] D. Andrijević, Semi-preopen sets, ibid. 38 (1986), 24–32.

[4] D. Andrijević, On the topology generated by preopen sets, ibid. 39 (1987), 367–376.

[5] D. Andrijević, On SP O-equivalent topologies, Suppl. Rend. Circ. Mat. Palermo 29 (1992), 317–328.

[6] H.H. Corson and E. Michael, Metrizability of certain countable unions, Illinois J. Math. 8 (1964), 351–360.

[7] M. Ganster and D. Andrijević, On some questions concerning semi-preopen sets, Journ. Inst. Math. and Comp. Sci. (Math. Ser.) 1 (1988), 65–75.

[8] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70 (1963), 36–41.

[9] A.S. Mashhour, M.E. Abd El-Monsef and S.N. El-Deeb, On precontinuous and weak precontinuous mappings, Proc. Math. Phys. Soc. Egypt 53 (1982), 47–53.

[10] O. Njåstad, On some classes of nearly open sets, Pacific J. Math. 15 (1965), 961–970.

(received 16.05.1996.)

Poljoprivredni fakultet, University of Belgrade, Nemanjina 6, 11081 Beograd—Zemun, Yugoslavia