A New Type of ξ-Open Sets Based on Operations

The aim of this paper is to introduce a new type of ξ-open sets in topological spaces which is called ξγ-open sets and we study some of their basic properties and characteristics.


Introduction
Ogata [9], introduced the concept of an operation on a topology, then after authors defined some other types of sets such as -open [9], -semi-open [6], -pre semi-open [6] and --open [1] sets in a topological space by using operations. In [4] the concept of -open set in a topological space is introduced and studied.
The purpose of this paper, is to introduce a new class of -open sets namely open sets and establish basic properties and relationships with other types of sets, also we define the notions of -neighbourhood, -derived, -closure and -interior of a set and give some of their properties which are mostly analogous to those properties of open sets. Throughout this paper, (X, ) or(briefly, X) mean a topological space on which no separation axioms are assumed unless explicitly stated. Let A be a subset of a topological space X, Cl(A) and Int(A) are denoted respectively the closure and interior of A.

Preliminaries.
We start this section by introducing some definitions and results concerning sets and spaces which will be used later. [7], if A Cl(Int(A)).
The complement of semi-open (resp., regular open, preopen and -open) set is said to be semi-closed (resp., regular closed, preclosed and -closed ).
Definition 2.2. [4] An open subset U of a space X is called -open if for each x U, there exists a semi-closed set F such that x F  U. The family of all -open subsets of a topological space (X, ) is denoted by O(X, ) or (briefly O(X)). The complement of each -open set is called -closed set. The family of all -closed subsets of a topological space (X, ) is denoted by C(X, ) or (briefly C(X)). Definition 2.3. [5] Let (X, ) be a topological space. An operation  on the topology  is a mapping from  into power set P(X) such that V  (V) for each V , where (V) denotes the value of  at V.  2) -closure of A is the intersection of all -closed sets containing A. Lemma 2.8. [4] 1) Let (Y, Y) be a subspace of (X, ). If F  SC(X, ) and F  Y, then F  SC(Y, Y).
2) Let Y be a subspace of a space X and Y  SC(X). If G  O(X) and G  Y, then G  O(Y).

 -Open Sets
In this section, a new class of -open sets called  -open sets in topological spaces is introduced. We define  to be a mapping on O(X) into P(X) and we say that : O(X) → P(X) is an -operation on O(X) if V  (V), for each V O(X).

Definition 3.1 A subset A of a space X is called  -open if for each point x A, there exist an
The family of all  -open subset of a topological space (X, ) is denoted by O(X, ) or (briefly O(X)).
A subset B of a space X is called  -closed if X \B is  -open. The family of all  -closed subsets of a topological space (X, ) is denoted by C(X, ) or (briefly C(X)).

Remark 3.2
From the definition of the operation , it is clear that (X)=X for any operation . For competence, it is assumed that ()=  for any -operation .

Remark 3.3 It is clear from the definition that every
 -open subset of a space X is open, but the converse is not true in general as shown in the following example:  The following example shows that the converse of the above proposition is not true in general.
The following example shows that the intersection of two -open sets need not be an -open set. Example 3.9 Consider X = {a, b, c} with discrete topology on X. Define an -operation  by From the above example, we notice that the family of all -open subsets of a space X is a supratopology and need not be a topology in general. If i I(Ai)  (i IAi) for any collection {Ai}i I  O(X), then  is said to be subadditive on O(X).      Here, we introduce some properties of -closure of the sets. Proof. They are obvious. In general, the equalities of (5) and (6) of the above proposition does not hold, as is shown in the following examples: Example 4.11 Consider X = {a, b, c} with discrete topology on X. Define an operation  on O(X) by    Proof. Assume that A is an -closed set and if possible that x is an -limit point of A which belongs to X \ A, then X \A is an -open set containing the -limit point of A, therefore, A  (X\A)  , which is contradiction. Conversely, assume that A is containing the set of its -limit points. For each x X\A, there exists an -open set U containing x such that A  U = , implies that x U  X\A, so by Proposition 3.10, X\A is an -open set hence, A is an -closed set.    Here, we introduce some properties of -Interior of the sets. Proof. Straightforward. In general, the equalities of (7) and (8)    The following two results can be easily proved.