On types of generalized closed sets

ABSTRACT The paper studies relations between types of generalized closed sets in topological spaces. It also answers an open question posed by Erdal.


Introduction and preliminaries
The concept of g-closedness of sets in topological spaces was first initiated by Levine [1] using the closure operator. Since then this concept has been investigated extensively by many topologists. They have started defining new types of generalized closed sets without paying attention that those sets happen to be equivalent. In this note, we work on those types of generalized closed sets that are equivalent. We also give answer to question of Erdal in [2, p. 265].
Throughout this paper, spaces mean topological spaces on which no any other property is assumed. For a subset A of a space X, the closure and interior of A with respect to X, respectively, are denoted by Cl x (A) and Int x (A) (or simply Cl(A) and Int(A)). Definition 1.1: A subset A of a space X is said to be (1) nowhere dense if Int(Cl(A)) = ∅, (2) preopen [3] if A # Int(Cl(A)), b-open [7] or sp-open [8] or γ-open [9] if A # Int(Cl(A)) < Cl(Int(A)).

Remark 1.2:
It is known that for any topological space (X, t), aO(X) # PO(X) < SO(X) # bO(X) # bO(X). Definition 1.3: Let X be a topological space and A # X. A point x [ X is said to be in the preclosure (resp. semi-closure, α-closure, β-closure, b-closure) of A if for every preopen (resp. semiopen, α-open, β- From Remark 1.2 and Definition 1.3, we have the following well-known lemma: Lemma 1.5: For any subset A of a space X, the following statements hold: (2) generalized semiclosed (briefly, gs-closed) [11] if weakly generalized closed (briefly, wg-closed) [20] if Cl(Int(A)) # U whenever A # U and U [ t.

The results
In this section, we start with the following lemma which will be used in the sequel. that hold for any subset of any topological space: All the authors who have introduced and defined sets in the above diagram claimed that none of the implications is reversible. But this turns to be false. Before giving the answer to what we have just said, we recall the following question posed by Erdal in [2].

Question 2.3:
Does there exist a subset of a space which is (a) bg-closed but not b-closed, (b) pg-closed or sg-closed but not bg-closed.
The next result provides the answer to above question and our claim.  That is Cl(X \ {x}) = X. Therefore Int(Cl(X \ {x})) = Int(X), since Int(X) = X and X \ {x} , X, then X \ {x} is preopen. Suppose for contraction that x Ó A. This implies that A # X \ {x}. By assumption, A is pgclosed, so p-Cl(A) # X \ {x}. Therefore x Ó p-Cl(A), which is a contradiction. Hence, x [ A. From (i) and (ii), we obtain that p-Cl(A) # A. Thus, A = p-Cl(A). This shows that A is preclosed.
Note that the same construction (as in (1)) can be applied to (2) and (3) with few modifications. But for the sake of completeness, we try to give these modifications to (2) and leave (3).
(2) (Only if part). Let A be a b-closed set. Then follow the same steps as in ( (1); only if part) and apply Fact 1.4 and Definition 1.6 (2). Hence A be bg-closed.
(If part). Let A be bg-closed. Since A # b-Cl(A) in general, it is enough to prove that b-Cl(A) # A. Let x [ b-Cl(A). By Lemma 2.1 , either {x} is preopen or nowhere dense. We have the following cases: Note that the above result has been proved by Mohammed et al. in [22] Lemma 2.6: Let (X, t) be a space. The following hold: Theorem 2.7: In a topological space (X, t), (1) every ga-closed set is pg-closed, (2) every sg-closed set is bg-closed, (3) every pg-closed set is bg-closed, (4) every bg-closed set is bg-closed.
(2) By Lemma 2.6 (2), every sg-closed set is b-closed and by Theorem 2.4 (2) b-closed set and bg-closed set are identical, hence the result.
On the other hand, if A is ag-closed, then α-Cl(A) , U whenever A # U and U [ t. By Lemma 2.12, Cl(Int(Cl(A))) # a-Cl(A) , U. Therefore, Cl(Int(Cl(A))) # U and so A isbg-closed.
(2) By the same way above using Lemma 2.12 (1) one can prove this. ▪ From all the above results, we try to give the final version of the diagram stated in Remark 2.2: Finally, we shall recall that Examples 23, 24 and 25 in [2], Example 2.5 in [25] and Example 3.21 in [19] show that none of the implications (in the middle column) are reversible.