1. Introduction

Semi-open sets were initially proposed by Levine [8] in 1963. In 1968,Velicko [13] was the first to allude to the class of -open subsets of a topological space. In 1979,Kasahara [7] developed the concept of α-closed graphs of functions and examined the idea of an operation on an open set τ. In 1982 Mashhour, Abd El-Monsef, and El-Deeb [9] developed the idea of preopen sets. Ogata [5] changed the designation of the operation α to operation on τ in 1991. In 1993, Raychaudhuri and Mukherjee [10] discovered and explored a class of sets called -preopen. Khalaf and Asaad [1] in 2009 introduced a new concept -open sets in topological spaces. Khalaf and Ameen [2] in 2010 introduced the new class of open sets called -open sets in topological spaces. Assad [3] introduced Operation approaches on -open sets and their separation axioms in 2016.Jayashree and Sivakamasundari [6] in 2018 initiated the Operation approaches on δ-open sets. An operation on the collection of -open subsets of a topological space was introduced by Asaad and Ameen [4] in 2019. Ameen, Asaad, and Muhammed [14] in 2019 introduced the super class of δ-open sets. In 2020, Vidhyapriya, Shanmugapriya and Sivakamsundari [12] developed a new type of open set called -open sets by combining the ideas of -preopen and -open sets. Shanmugapriya, Vidhyapriya and Sivakamsundari [11] in 2021 introduced a -open set as a novel class of operational open sets in topological spaces. Using -open sets, here we examined fundamental properties of topological operators: the -closure, , -limitpoints,-derivedsets,-neighborhood, -interior,-exterior, -boundary,-frontier of a set and -saturated sets.

2. Preliminaries

Let be a set and be topology then be the topological space. Here be a subset of the topological space .

Definition 2.1[8]. A subset is called semi-open if .

The complement is called a semi-closed set.

Definition 2.2[12].

a. A -preopen subset of a space is called a -open set if for all , ∃ a semi-closed set F such that .The collection of all -open sets is denoted by .

b. The point in is called -closure of iff , for every -open set containing .It is denoted by

Definition 2.2[11]. An operation on is a mapping such that for every , where is the power set of and is the value of under .

Definition 2.3[11]. Let be a topological space and be an operation on . A nonempty set of is called a -open set if for all , ∃ a -open set such that and

Proposition 2.4[11]. Every -open set is a -open set.

Proposition 2.3[11]. The union of any class of -open sets in is -open.

Definition 2.5[11]. Let be any topological space. An operation on is said to be -operation if for each and for every pair of -open sets H₁ and H₂ such that both containing , ∃ a - open set F containing such that .

Definition 2.6[11]. A topological space (Y, τ) with an operation on is said to be - regular space if for given and for each -open set H containing , ∃ a -open set F containing such that .

3. TOPOLOGICAL PROPERTIES OF -OPEN SETS

3.1. AND -CLOSURE OF A SET

Definition 3.1.1. Let be a subset of a topological space and be an operation on . A point is called point of the set if for all-open set H containing , .The family of points of is called of and is denoted by .

Proposition 3.1.2. Let be a subset of a topological space and be an operation on , then .

Proof. Consider let then by Definition 2.2 , for every -open set containing . Always , Which gives . By Definition 3.1.1, .

Thus,.

Definition 3.1.3. Let be a subset of a topological space and be an operation on . The -closure of is defined as the intersection of all -closed sets of containing and it is denoted by .

-open set in

Proposition 3.1.4. Let and be subsets of a topological space and be an operation on , then the following statements are true:

a. .

b. .

c. is a -closed set in and it is the smallest -closed set containing .

d. and .

e. is a -closed set iff .

f. is a -closed set iff .

g. If , then and .

h. .

i. .

j. .

k. .

l. .

Proposition 3.1.5. If is a -regular space, then .

Proof. is proved in Proposition 3.1.2.

Let , then ∃ a -open set H containing p such that . As is a -regular space then by Definition 3.1.4, for all and for all -open set containing such that , so . Hence.Therefore .

Proposition 3.1.6. If is a -operation on , then

b. .

Proof. (a) From Proposition 3.1.4 (f) we have, .

On the other side, consider ∃ a pair of -open sets H₁ and H₂ such that both containing , and . Now is a -operation on then for all and ∃ a -open set F containing such that .

So,

but

.Which gives, .

Yields, .

So, .

Thus, .

(b) From Proposition 3.1.4 (k)

so it is enough to obtain that.

Consider, then there exist two -open sets and containing such that = and = . As is a -operation on then by Proposition 3.1.2 is a -open set in Y so = . Finally , thus

Proposition 3.1.7. Let be a subset of a topological space and be an operation on . Then , for every -open set of containing .

Proof. Consider and suppose for some -open set of containing .

Then and is a -closed set in Y. Then, . Thus, , this is a contradiction. So for every - open set G of Y containing .

To prove the contrary, if then ∃ a -closed set E such that and .

Then is a -open set such that contradicting our hypothesis.

Thus, .

3.2. -LIMIT POINT

Definition 3.2.1. Let P be a subset of a topological space and be an operation on . A point is called - limit point of P if for every -open set G containing , .

The family of - limit points of P is called a -derived set of P and it is denoted by .

Some properties of -derived set are mentioned in the following propositions.

Proposition 3.2.2. The following properties hold for any sets P and Q in a topological space with an operation on.

b. If , then.

c. .

d. .

e. .

f. .

Proof. Proof of (a) is obvious.

(b)We have , .Then

Hence .

Thus

This example proves that the reverse inequality of (c),(d), (e) and (f) is not true in general.

Example 3.2.3.Consider the space and and

.An operation is defined as follows, for every .

Implies .

For a subset and therefore,

Hence the reverse inequality of (c) is not true.

For a subset and therefore,

Hence the reverse inequality of (d) is not true.

For a subset therefore,

Hence the reverse inequality of (e) is not true.

For a subset therefore,

Hence the reverse inequality of (f) is not true.

Corollary 3.2.4. Let P be a subset of a topological space and be an operation on . Then

Proof. Let from Definition 3.2.1, for every -open set containing satisfying

, it follows that . So, the corollary is proved.

3.3.-NEIGHBOURHOOD OF A POINT AND A SET

Definition 3.3.1. A subset N of a topological space is called a -neighbourhood of a point , if ∃ a -open set G in Y such that .

The collection of all -neighbourhood is denoted by

Definition 3.3.2. A subset N of a topological space is called a -neighbourhood of a set if ∃ a -open set G in Y such that .

Remark 3.3.3. Let be a -open set iff it is a -neighbourhood of each of its points.

Proposition 3.3.4. If E is a -closed subset of a topological space Y and , then ∃ a -neighbourhood N of such that

Proof. Let E be -closed subset of a topological space Y, then is-open set. Let . By Remark 3.3.3, is a -neighbourhood of each of its points. Finally ∃ a -neighbourhood N of such that

which gives that .

Proposition 3.3.5. For a topological space the following results of -neighbourhood are true for all in Y:

a..

b. If , then .

c. If and , then .

d. If and , then .

e. If then there exist such that and for all .

Proof. (a) Since by definition of -neighbourhood,

which is a -open set of such that implies is in for all .Therefore.

(b) Let implies is -neighbourhood of , then by definition .

, implies is a -neighbourhood of , that is .

(d) Let and , then there exists two -open sets and such that

and . Then . Since is a -open set we get is a -neighbourhood of. Thus, .

(e) Let , then there exist a -open set such that .Since is a -open set, it is -neighbourhood of each of its points by Remark 3.3.3.Thus, there exist a -open set such that and for all .

3.4.-INTERIOR

Definition 3.4.1. Let P be a subset of a topological space and be an operation on A point is called -interior point of if ∃ a-open set H containing , such that and we denote the family of such points by .

Proposition 3.4.2. Let be an operation on , then for a subset of Y

and.

Proof. Let , then by Definition 3.4.1, ∃ a -open set such that and . .

Let , which is a -open set by Proposition 2.3 and Proposition 2.5. Finally, .

So and.

Proposition 3.4.3. Let be an operation on , then the following properties holds:

a. and

b. is the largest -open set contained in .

c. is -open set iff .

d. If , then .

e. .

f. .

g. .

Proof. Proof of (a), (b), (c) and (g) are obvious from existing results and definitions.

The following example shows that the reverse inequality of (d), (e) and (f) is not true in general.

Example 3.4.4.Consider the space and . An operation is defined as follows, for every

Implies .

For a subset and and . Hence .

Hence the reverse inequality of (d) is not true.

For a subset and , implies .

Hence,

Thus, the reverse inequality of (e) is not true.

For a subset and , implies .

Hence,.

Hence the reverse inequality of (f) is not true.

Proposition 3.4.5. If is a -operation on then for part of subsets of a space Y, .

Proof. We have by Proposition 3.4.3(f).

On the other hand, let implies that, and.Then there exists-open sets and containing such that and , so .Since is a-operation, there exists a -open set containing such that , hence .