Insertion of a Contra-continuous Function between Two Comparable Real-valued Functions

A necessary and sufficient condition in terms of lower cut sets are given for the 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 C-open set in a topological space was introduced by Hatir et al. [12]. The authors define a set S to be a C-open set if , where U is open and A is semi-preclosed. A set S is a C-closed set if its complement (denoted by ) In [7] it was shown that a set A is β-open if and only if ( ) ( ) ( ). 23 If g and f are real-valued functions defined on a space X, we write f g ≤ ( ) for all x in X.
The following definitions are modifications of conditions considered in [17].
A property P defined relative to a real-valued function on a topological space is a cc-property 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 and 2 P are cc-properties, the following terminology is used: (i) A space X has the weak cc-insertion property for ( ) 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 a necessary and sufficient condition for the space to have the 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 between two comparable real-valued functions has also recently considered by the authors in [20,21].

The Main Result
Before giving a sufficient condition for insertability of a contra-continuous function, the necessary definitions and terminology are stated. , τ X We define the subsets Λ A and V A as follows: In [6,18,22], Λ A is called the kernel of A.
The family of all α-open, α-closed, C-open and C-closed will be denoted by , τ X CC respectively.
The following first two definitions are modifications of conditions considered in [15,16].
The concept of a lower indefinite cut set for a real-valued function was defined by Brooks [2] as follows: for a real number , We now give the following main result: Let g and f be real-valued functions defined on the X such that . , , Define functions F and G mapping the rational numbers Q into the power set of X If 1 t and 2 t are any elements of Q with , By Lemmas 1 and 2 in [16] 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 , , h is a contra-continuous function on X.
The above proof used the technique of Theorem 1 in [15].
Theorem 2.2. Let 1 P and 2 P be cc-property and X be a space that satisfies the weak cc-insertion property for ( ).
, 2 1 P P Also assume that g and f are functions on X such that , f g < g has property 1 P and f has property . Proof. Assume that X has the weak cc-insertion property for ( ).
, 2 1 P P Let g and f be functions such that , f g < g has property 1 P and f has property .   x then .
Since 1 P and 2 P are E-properties, then 1 g has property 1 P and 1 f has property . 2 P Since X has the weak cc-insertion property for ( ), , 2 1 P P then there exists a contra-continuous function h such that .
it follows that X satisfies the cc-insertion property for ( ).
Conversely, let g and f be functions on X such that g has property , 1 P f has property 2 P and . f g < By hypothesis, there exists a contra-continuous function h such that We follow an idea contained in Lane [17]. Since the constant function 0 has property , 1 P since h f − has property , 2 P and since X has the cc-insertion property for ( ), 3 :

Applications
The abbreviations cαc and cCc are used for contra-α-continuous and contra-Ccontinuous, respectively.
Before stating the consequences of Theorems 2.1, 2.2, we suppose that X is a topological space whose kernel sets are open. ).

Proof.
Let g and f be real-valued functions defined on X, such that f and g are c cα ( ), . resp cCc 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 , : , , The proof follows from Theorem 2.

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

Proof.
Let g and f be real-valued functions defined on X, such that g is cαc (resp. 29 cCc) and f is cCc (resp. cαc), 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 , , : : , : , , Furthermore, by definition, It remains only to prove that h is a contra-continuous function on X. For we obtain a decreasing sequence of closed subsets of X with the required properties.
is a countable covering of C-closed (resp. α-closed) subsets of X, we set for , , . resp c c cCc α □