Topological-like notions via Δ -open sets

We provide a characterization of Δ -open and Δ -closed sets in topological spaces. Besides, based on the concepts of Δ -open and Δ -closed sets we investigate the notions of Δ -interior, Δ -exterior, Δ -closure, Δ -derived sets, Δ -boundary sets


Introduction
Several notions of open-like and closed-like sets in topological spaces were introduced and studied.The beginning was with Norman Levine who initiated the notion of semi open sets, [12].After that α-sets and β-sets in topological spaces were introduced and considered as "nearly open" by Olav Njåstad, [17].Furthermore, the notions of θ-open and δ-open sets were initiated, [24].Later the concepts of pre-open sets and semi-preopen sets were started, [15] and [6], respectively.
The concept of closed sets in topological spaces was extended to generalized closed sets, [13].However, another extension of closed sets in topological spaces called semi generalized closed sets was obtained in [7].N. Palaniappan and K. Chandrasekhara Rao introduced regular generalized closed sets, [19].Additionally, a class of closed-like sets called ψ-closed sets was established by M. Veera Kumar, [23].
A set in a topological space is called Δ-open if it is the symmetric difference of two open sets.The notion of Δ-open sets appeared in [18] and in [10].However, it was pointed out in [18] and in [10] that the notion of Δ-open sets is due to a preprint by M. Veera Kumar.The complement of a Δ-open set is Δ-closed.
In Section 2, we establish a characterization for Δ-open (respectively, Δ-closed) sets that is free of the symmetric difference operation, Theorem 2.2 (respectively, Corollary 2.3).As a consequence, we see that a finite intersection (union In Section 3, notions of Δ-interior, Δ-exterior, and Δ-closure are investigated.Characterizations and properties of these notions are introduced as well.
Concepts and properties of Δ-limit points, Δ-boundary points, and Δ-dense sets are presented in Sections 4 and 5.
Section 6 is devoted to examine Δ-open and Δ-closed sets in the product topology.It is shown that a product of two Δ-open sets is again a Δ-open in the product topology, Theorem 6.2.However, this is not the case for Δ-closed sets as illustrated by Example 6.5.
Finally in Section 7 we consider Δ-open sets, Δ-closed sets, Δ-interior, and Δ-closure in subspace topology.In particular, characterizations of Δ-open and Δ-closed sets in subspace topology are given, Propositions 7.1 and 7.2.

Δ-open and Δ-closed sets
In this section we provide basic notions and results related to Δ-open and Δ-closed sets.These results will be used and applied in the subsequent sections.
Recall that for sets A and B their symmetric difference is given as   , , , , , , , , , ,  , .
Clearly, the sets { , , } a b c and { } e are Δ-open, whereas their union is not.It is worth noting from the previous example that the collection of all Δ-open sets does not form a topology in general.

Then considering  under the standard topology, each S n is open set and so it is
It should be pointed out that the open set O and the closed set C in Corollary 2.3 can be chosen to be disjoint.
A topological space is said to be T 1 2 if each g-closed set is closed, [13].The T 1

2
-spaces were first defined by Norman Levine, [13].Hereinafter, William Dunham obtained an interesting characteriza- The converse of Proposition 2.6 need not be true.

Let ( , )
X τ be a topological space and A X ⊆ .Recall that the union of all open sets contained in A is known as the interior of A and is is denoted by Int A ( ).The exterior of a set is the interior of it complement, and it is denoted by Ext A ( ).Besides, the intersection of all closed sets containing A is known as the closure of A and is is denoted by Cl A ( ).Likewise we have the following notions.

(i). The union of all Δ-open sets contained in A is said to be the Δ-interior of A and is denoted by
Because each open set is Δ-open and each closed set is Δ-closed, the following result follows directly.
Lemma 3.3.Let ( , ) X τ be a topological space and A X ⊆ .Then We state and prove the following result on a relation between ΔInt A ( ) and ΔCl A ( ).
A combination of Lemma 3.3 and Proposition 3.4 yields the next result.
The proof of the following result is trivial and so it is omitted.X τ be a topological space and A B X , ⊆ .Then Proof. ( Fundamental properties of Δ-closure are given below. Proposition 3.13.Let ( , ) X τ be a topological space and A B X , ⊆ .Then Proof.     .So Lemma 6.1(i) implies that

Δ-open sets in subspace topology
Let X ,W be a topological space and Y X ⊆ .Then it is known that the collection X τ be a topological space, Y X ⊆ , and X τ be a topological space, Y X ⊆ , and

Conclusion
In this paper we provide characterizations for Δ-open sets and Δ-closed sets that are independent of the symmetric difference operation; Theorem 2.2 and Corollary 2.3.We define and investigate the notions of Δ-interior, Δ-closure, Δ-limit points, Δ-boundary points, and Δ-dense sets.The investigation continues to Δ-open and Δ-closed sets in product topology and in subspace topology.Many of the properties of those "Δ notions" agree with their corresponding topological notions.However, several counter examples are given to show distinction between the "Δ notions" and the usual topological notions.
For a future work we are going to define and investigate the concepts of Δ-continuous functions and Δ-irresolute functions in topological spaces.
The collection of all Δ-open sets in X is W I 'o X a b c a b c d d e d e ^`^`^`^`^,

1 ∪
be an enumeration of the rationals.For each n ∈ ,

Let's consider Example 2 . 4 . 2 - 2 -
For A a b = { , } and B c d = { , }, we have A B I and so 'Cl A B that in statement (iii) of Proposition 3.13 the inclusion is strict.The next result points out that in a T 1 space, each set and its Δ-closure are the same.Proposition 3.14.Let ( , ) X τ be a T 1 space and A X ⊆ .Then 'Cl A A ( ) .

Example 4 . 4
Let X a b c d e = { , , , , } with a topology c d a b c d Let W 'o denote the collection of all Δ-open sets in X.Then W I 'o X a b c d a b c d a b c e d e e ^`^`^`^`^`^, , ,

1 2 × 1 2.
. Due to Lemma 6.1(ii), we have B B B u In spite of the Cartesian product of two Δ-closed sets B In the same manner, it follows from Corollary 2.3 that any open set and any closed set is Δ-closed.Additionally, a finite union of Δ-closed sets is Δ-closed.Nonetheless, intersection of two Δ-closed sets need not be Δ-closed and arbitrary union of Δ-closed sets need not be Δ-closed.
[10]nition 2.1.([18]and[10])A set A in a topological space ( , ) X τ is called Δ-open if there are open sets O 1 and O 2 so that A O O 1 2 For an open set O in a topological space ( , ) X τ , it is obvious that O O 'I , so every open set is Δ-' is Δ-open set which is not open in the standard topology on  .The complement of a Δ-open set is called Δ-closed.We provide a characterization of Δ-open sets.Theorem 2.2.A set A in a topological space ( , ) X τ is Δ-open if and only if there are an open The intersection of all Δ-closed sets containing A is said to be the Δ-closure of A and is denoted by ΔCl A ( ).
Clearly, ΔInt A ( ) need not be Δ-open and ΔCl A ( ) need not be Δ-closed.It should be also noted that ifA is Δ-open, then 'Int A A ( ), and if A is Δ-closed, then 'Cl A A ( ).In either case the converse is not true.
Basic properties of Δ-interior are summarized in the next proposition.
Example 2.8 confirms that the converse of Proposition 3.10 is not correct.In the same manner we give a characterization of Δ-closure in terms of Δ-neighborhoods.For the backward direction, suppose that for any ΔN x ( ).Then there are Δ-neighborhoods of x, ΔN x 1 ( ) and ΔN x Then Theorem 2.2 assures that there are an open set O 1 and a closed set C 1 in X 1 , and there are an open set O 2 and a closed set C 2 in X 2 , such that A O C be two topological spaces.Let B 1 be Δ-closed in X 1 and B 2 be Δ-closed in X 2 .Then Corollary 2.3 guarantees that there are an open set O 1 and a closed set C 1 in X 1 , and there are an open set O 2 and a closed set C 2 in X 2 , such that B O C open in forms a topology on Y, named as the subspace topology.Recall that a set B is closed in Y if and only if there is a closed set C in X satisfying B Y C .We aim in this section to study Δ-open and Δ-closed sets in the subspace topology.⊆ .Then, S is Δ-open in Y if and only if there is a Δ-open set A in X such that S Y A .Proof.Let S Y ⊆ .Then, S is Δ-open in Y if and only if there is an open set U in Y and a closed set K in Y such that S U K if and only if there is an open set O in X and a closed set C in X such that S is Δ-closed in Y if and only if there is an open set U in Y and a closed set K in Y such that S U K if and only if there is an open set O in X and a closed set C in X such Then Proposition 7.2 implies that there is a Δ-closed set B in X such that 'Cl A Y B Generally the inclusion in Proposition 7.5 is strict.With reference to Example 2.4, let Y a b d = { , , } and A a b = { , }.Then 'Int A ( ) I but 'Int A A Y ( ) is Δ-closed in Y. Y ( ) .