Strong Insertion of a Contra-continuous Function between Two Comparable Contra-precontinuous (Contra-semi-continuous) of Real-valued Functions

Necessary and sufficient conditions in terms of lower cut sets are given for the strong insertion of a contra-continuous function between two comparable real-valued functions on such topological spaces that kernel of sets are open.


Introduction
The concept of a preopen set in a topological space was introduced by Corson and Michael in 1964 [4]. A subset A of a topological space ( ) A generalized class of closed sets was considered by Maki in [20]. He investigated the sets that can be represented as union of closed sets and called them V-sets. Complements of V-sets, i.e., sets that are intersection of open sets are called Λ-sets [20].
Recall that a real-valued function f defined on a topological space X is called Acontinuous [28] if the preimage of every open subset of R belongs to A, where A is a collection of subsets of X. Most of the definitions of function used throughout this paper are consequences of the definition of A-continuity. However, for unknown concepts the reader may refer to [5,11]. In the recent literature many topologists had focused their research in the direction of investigating different types of generalized continuity.
Hence, a real-valued function f defined on a topological space X is called contracontinuous (resp. contra-semi-continuous, contra-precontinuous) if the preimage of every open subset of R is closed (resp. semi-closed, pre-closed) in X [6].
Results of Katětov [14,15] concerning binary relations and the concept of an indefinite lower cut set for a real-valued function, which is due to Brooks [2], are used in order to give a necessary and sufficient conditions for the insertion of a contracontinuous function between two comparable real-valued functions on such topological spaces that Λ-sets or kernel of sets are open [20].
If g and f are real-valued functions defined on a space X, we write The following definitions are modifications of conditions considered in [16].
A property P defined relative to a real-valued function on a topological space is a ccproperty provided that any constant function has property P and provided that the sum of a function with property P and any contra-continuous function also has property P. If 1 P In this paper, for a topological space whose Λ-sets or kernel of sets are open, is given a sufficient condition for the weak cc-insertion property. Also for a space with the weak cc-insertion property, we give necessary and sufficient conditions for the space to have the strong cc-insertion property. Several insertion theorems are obtained as corollaries of these results. In addition, the insertion and strong insertion of a contra-α-continuous function and insertion of a contra-continuous function between two comparable realvalued functions has also recently considered by the authors in [22,23,24].

The Main Result
Before giving a sufficient condition for insertability of a contra-continuous function, the necessary definitions and terminology are stated.
The abbreviations cc, cpc and csc are used for contra-continuous, contraprecontinuous and contra-semi-continuous, respectively. Definition 2.1. Let A be a subset of a topological space ( ).
, τ X We define the subsets Λ A and V A as follows: The family of all preopen, preclosed, semi-open and semi-closed will be denoted by , τ X sC respectively.
We define the subsets ( ), The following first two definitions are modifications of conditions considered in [14,15].  ..., , The concept of a lower indefinite cut set for a real-valued function was defined by Brooks [2] as follows: for a real number , is called a lower indefinite cut set in the domain of f at the level . ℓ We now give the following main result: Define functions F and G mapping the rational numbers Q into the power set of X By Lemmas 1 and 2 of [15] it follows that there exists a function H mapping Q into the power set of X such that if 1 t and 2 t are any rational numbers with , We first verify that Also, for any rational numbers 1 t and 2 t with , The above proof used the technique of Theorem 1 in [14].
If a space has the strong cc-insertion property for ( ), , 2 1 P P then it has the weak cc-insertion property for ( ).
, 2 1 P P The following result uses lower cut sets and gives a necessary and sufficient condition for a space satisfies that weak cc-insertion property to satisfy the strong cc-insertion property.
are completely separated by contra-continuous functions, and (ii) of lower cut sets for for each n and hence then there exists an n such that n F x ∈ and hence ( ) and this verifies (1). Next, in order to establish (2), note Since 1 P and 2 P are cc-properties, then 1 g has property 1 P and 1 f has property . 2

P
Since by hypothesis X has the weak cc-insertion property for ( ), , 2 1 P P then there exists a contra-continuous function h such that .
Therefore X has the strong cc-insertion property for ( ).
, 2 1 P P (The technique of this proof is by Lane [16].) Conversely, assume that X satisfies the strong cc-insertion for ( ).
We follow an idea contained in Powderly [27]. Now consider the functions 0 and . h f − 0 satisfies property 1 P and h f − satisfies property . 2 P Thus there exists a contra-continuous function 1 h for any x in X, then ( ) ( ) ( ).
On the other hand, if We next show that n k is a contra-continuous function which completely separates n F and ( ).
Hence X satisfies the strong cc-insertion property for ( ).

, 2 1 P P
The converse is obvious since any contra-continuous function must satisfy both properties 1 P and .
2 P (The technique of this proof is by Lane [17].) □

Applications
Before stating the consequences of Theorems 2.1, 2.2, and 2.3 we suppose that X is a topological space whose kernel sets are open. ( ) csc csc, ).

Proof.
Let g and f be real-valued functions defined on X, such that f and g are cpc (resp. csc), and . f g ≤ If a binary relation ρ is defined by , then by hypothesis ρ is a strong binary relation in the power set of X. If 1 t and 2 t are any elements of Q with , : is a preopen (resp. semi-open) set and since is a preclosed (resp. semi-closed) set, it follows that ).

Proof.
Let g and f be real-valued functions defined on the X, such that f and g are cpc (resp. csc), and .

Proof.
Let g and f be real-valued functions defined on X, such that g is cpc (resp. csc) and f is csc (resp. cpc), with . f g ≤ If a binary relation ρ is defined by , then by hypothesis ρ is a strong binary relation in the power set of X. If 1 t and 2 t are any elements of Q with , , : : , : is a semi-open (resp. preopen) set and since is a preclosed (resp. semi-closed) set, it follows that  G is a semi-open (resp. preopen) subset of X which is contained in a preclosed (resp. semi-closed) subset F of X, then there exists a closed subset H of X such that Furthermore, by definition, It remains only to prove that h is a contra-continuous function on X. For  (ii) Every preopen (resp. semi-open) subset of X is a closed subsets of X.
(iii) Every preclosed (resp. semi-closed) subset of X is an open subsets of X.   , cpc csc Now, assume that g and f are functions on X such that , f g ≤ g is cpc (resp. csc) and f is cc. Since g f − is cpc (resp. csc), therefore the lower cut set is a preopen (resp. semi-open) subset of X.