Strong insertion of a contra-continuous function between two comparable 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. M.S.C. 2010: Primary 54C08, 54C10, 54C50; Secondary 26A15, 54C30.


Introduction
The concept of a C−open set in a topological space was introduced by E. Hatir, T. Noiri and S. Yksel in [12]. In [7] it was shown that a set A is β−open if and only if A ⊆ Cl(Int(Cl(A))). 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 A−continuous [25] 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 [4,11]. In the recent literature many topologists had focused their research in the direction of investigating different types of generalized continuity.
J. Dontchev in [5] introduced a new class of mappings called contra-continuity.S. Jafari and T. Noiri in [13,14] exhibited and studied among others a new weaker form of this class of mappings called contra-α−continuous. A good number of researchers have also initiated different types of contra-continuous like mappings in the papers [1,3,8,9,10,23].
Hence, a real-valued function f defined on a topological space X is called contracontinuous (resp. contra-C−continuous , contra-α−continuous) if the preimage of every open subset of R is closed (resp. C−closed , α−closed) in X [5].
Results of Katětov [15,16] 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 g ≤ f in case 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 P 1 and P 2 are cc−properties, the following terminology is used:(i) A space X has the weak cc−insertion property for (P 1 , P 2 ) if and only if for any functions g and f on X such that g ≤ f, g has property P 1 and f has property P 2 , then there exists a contra-continuous function h such that g ≤ h ≤ f .(ii) A space X has the strong cc−insertion property for (P 1 , P 2 ) if and only if for any functions g and f on X such that g ≤ f, g has property P 1 and f has property P 2 , then there exists a contra-continuous function h such that g ≤ h ≤ f and if g(x) < f (x) for any x in X, then g(x) < h(x) < f (x).
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 of a contra-continuous function between two comparable contra-precontinuous real-valued functions has also recently considered by the author in [21].

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, cαc and cCc are used for contra-continuous, contra-α−continuous and contra-C−continuous, respectively. Definition 2.1. Let A be a subset of a topological space (X, τ ). We define the subsets A Λ and A V as follows: In [6,19,22], A Λ is called the kernel of A.
We define the subsets α(A Λ ), α(A V ), C(A Λ ) and C(A V ) as follows: The following first two definitions are modifications of conditions considered in [15,16].

Definition 2.2.
If ρ is a binary relation in a set S thenρ is defined as follows: xρ y if and only if y ρ v implies x ρ v and u ρ x implies u ρ y for any u and v in S.

Definition 2.3.
A binary relation ρ in the power set P (X) of a topological space X is called a strong binary relation in P (X) in case ρ satisfies each of the following conditions: 1) If A i ρ B j for any i ∈ {1, . . . , m} and for any j ∈ {1, . . . , n}, then there exists a set C in P (X) such that A i ρ C and C ρ B j for any i ∈ {1, . . . , m} and any j ∈ {1, . . . , n}. 2 The concept of a lower indefinite cut set for a real-valued function was defined by Brooks [2] as follows: We now give the following main result: in the domain of f and g at the level t for each rational number t such that if t 1 < t 2 then A(f, t 1 ) ρ A(g, t 2 ), then there exists a contra-continuous function h defined on X such that g ≤ h ≤ f .

Proof.
Let g and f be real-valued functions defined on the X such that g ≤ f . By hypothesis there exists a strong binary relation ρ on the power set of X and there exist lower indefinite cut sets A(f, t) and A (g, t) in the domain of f and g at the level t for each rational number t such that if t 1 < t 2 then A(f, t 1 ) ρ A(g, t 2 ). Define functions F and G mapping the rational numbers Q into the power set of X by , and F (t 1 ) ρ G(t 2 ). By Lemmas 1 and 2 of [16] it follows that there exists a function H mapping Q into the power set of X such that if t 1 and t 2 are any rational numbers with For Also, for any rational numbers t 1 and t 2 with The above proof used the technique of theorem 1 in [15].
If a space has the strong cc-insertion property for (P 1 , P 2 ), then it has the weak cc-insertion property for (P 1 , P 2 ).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.
Theorem 2.2. Let P 1 and P 2 be cc−property and X be a space that satisfies the weak cc−insertion property for (P 1 , P 2 ). Also assume that g and f are functions on X such that g ≤ f, g has property P 1 and f has property P 2 . The space X has the strong cc−insertion property for (P 1 , P 2 ) if and only if there exist lower cut sets A(f − g, 2 −n ) and there exists a sequence {F n } of subsets of X such that (i) for each n, F n and A(f − g, 2 −n ) are completely separated by contra-continuous functions, and Suppose that there is a sequence (A(f − g, 2 −n )) of lower cut sets for f − g and suppose that there is a sequence (F n ) of subsets of X such that F n and such that for each n, there exists a contra-continuous function k n on X into [0, 2 −n ] with k n = 2 −n on F n and k n = 0 on A(f − g, 2 −n ). The function k from X into [0, 1/4] which is defined by is a contra-continuous function by the Cauchy condition and the properties of contracontinuous functions, for each n and hence k n (x) = 0 for each n. Thus k(x) = 0.
Conversely, if (f − g)(x) > 0, then there exists an n such that x ∈ F n and hence k n (x) = 2 −n . Thus k(x) ̸ = 0 and this verifies (1). Next, in order to establish (2), note that In the former case, and in the latter, Since P 1 and P 2 are cc−properties, then g 1 has property P 1 and f 1 has property P 2 . Since by hypothesis X has the weak cc−insertion property for (P 1 , P 2 ) , then there exists a contra-continuous function h such that . Therefore X has the strong cc−insertion property for (P 1 , P 2 ). (The technique of this proof is by Lane [17].) Conversely, assume that X satisfies the strong cc−insertion for (P 1 , P 2 ). Let g and f be functions on X satisfying P 1 and P 2 respectively such that g ≤ f . Thus there exists a contra-continuous function h such that g ≤ h ≤ f and such that if g(x) < f (x) for any x in X, then g(x) < h(x) < f (x). We follow an idea contained in Powderly [24]. Now consider the functions 0 and f − h.0 satisfies property P 1 and f − h satisfies property P 2 . Thus there exists a contra-continuous function h 1 such We next show that k n is a contra-continuous function which completely separates F n and A(f − g, 2 −n ). From its definition and by the properties of contra-continuous functions, it is clear that k n is a contra-continuous function. Let x ∈ F n . Then, from the definition of k n , k n ( Thus k n (x) = 0, according to the definition of k n . Hence k n completely separates F n and A(f − g, 2 −n ). Theorem 2.3. Let P 1 and P 2 be cc−properties and assume that the space X satisfied the weak cc−insertion property for (P 1 , P 2 ). The space X satisfies the strong cc−insertion property for (P 1 , P 2 ) if and only if X satisfies the strong cc−insertion property for (P 1 , cc) and for (cc, P 2 ). Proof. Assume that X satisfies the strong cc−insertion property for (P 1 , cc) and for (cc, P 2 ). If g and f are functions on X such that g ≤ f, g satisfies property P 1 , and f satisfies property P 2 , then since X satisfies the weak cc−insertion property for (P 1 , P 2 ) there is a contra-continuous function k such that g ≤ k ≤ f . Also, by hypothesis there exist contra-continuous functions h 1 and h 2 such that g ≤ h 1 ≤ k and if g( Hence X satisfies the strong cc−insertion property for (P 1 , P 2 ).
The converse is obvious since any contra-continuous function must satisfy both properties P 1 and P 2 . (The technique of this proof is by Lane [18].)

Applications
Before stating the consequences of Theorems 2.1, 2.2 and 2.3 we suppose that X is a topological space whose kernel of sets are open.  C−open) sets G 1 , G 2 of X , there exist closed sets F 1 and F 2 of X such that G 1 ⊆ F 1 , G 2 ⊆ F 2 and F 1 ∩ F 2 = ∅ then X has the weak cc−insertion property for (cαc, cαc) (resp. (cCc, cCc)). Proof. Let g and f be real-valued functions defined on X, such that f and g are cαc (resp. cCc), and g ≤ f .If a binary relation ρ is defined by A ρ B in case α(A Λ ) ⊆ α(B V ) (resp. C(A Λ ) ⊆ C(B V )), then by hypothesis ρ is a strong binary relation in the power set of X. If t 1 and t 2 are any elements of Q with t 1 < t 2 , then A(g, t 2 ). The proof follows from Theorem 2.1.  C−open) sets G 1 , G 2 , there exist closed sets F 1 and F 2 such that G 1 ⊆ F 1 , G 2 ⊆ F 2 and F 1 ∩ F 2 = ∅ then every contra-α−continuous (resp. contra-C−continuous) function is contra-continuous.
Proof. Let f be a real-valued contra-α−continuous (resp. contra-C−continuous) function defined on X. Set g = f , then by Corollary 3.1, there exists a contracontinuous function h such that g = h = f .  C−open) sets G 1 , G 2 of X , there exist closed sets F 1 and F 2 of X such that G 1 ⊆ F 1 , G 2 ⊆ F 2 and F 1 ∩ F 2 = ∅ then X has the strong cc−insertion property for (cαc, cαc) (resp. (cCc, cCc)). Proof. Let g and f be real-valued functions defined on the X, such that f and g are cαc (resp. cCc), and g ≤ f . Set h = (f + g)/2, thus g ≤ h ≤ f and if g(x) < f (x) for any x in X, then g(x) < h(x) < f (x). Also, by Corollary 3.2, since g and f are contra-continuous functions hence h is a contra-continuous function.

Corollary 3.4.
If for each pair of disjoint subsets G 1 , G 2 of X , such that G 1 is α−open and G 2 is C−open, there exist closed subsets F 1 and F 2 of X such that G 1 ⊆ F 1 , G 2 ⊆ F 2 and F 1 ∩ F 2 = ∅ then X have the weak cc−insertion property for (cαc, cCc) and (cCc, cαc). Proof. Let g and f be real-valued functions defined on X, such that g is cαc (resp. cCc) and f is cCc (resp. cαc), with g ≤ f .If a binary relation ρ is defined by , then by hypothesis ρ is a strong binary relation in the power set of X. If t 1 and t 2 are any elements of Q with t 1 < t 2 , then The proof follows from Theorem 2.1.
Before stating consequences of Theorem 2.2, we state and prove the necessary lemmas. Lemma 3.1. The following conditions on the space X are equivalent: (i) For each pair of disjoint subsets G 1 , By (i) there exists two disjoint closed subsets of X, and G 1 (resp. G 2 ) is a closed subsets of X, then there exists a contra-continuous function h : Proof. Suppose that G 1 (resp. G 2 ) is a closed subset of X. By Lemma 3.2, there exists a contra-continuous function h : Then for every n ∈ N, F n is a closed subset of X and (ii) ⇔ (iii) By De Morgan law and noting that the complement of every open subset of X is a closed subset of X and complement of every closed subset of X is an open subset of X, the equivalence is hold. Then F 1 and F 2 are two disjoint closed subsets of X that contain G 1 and G 2 , respectively. Hence by Corollary 3.4, X has the weak cc−insertion property for (cαc, cCc) and (cCc, cαc). Now, assume that g and f are functions on X such that g ≤ f, g is cαc (resp. cCc) and f is cc. Since f − g is cαc (resp. cCc), therefore the lower cut set A(f − g, 2 −n ) = {x ∈ X : (f − g)(x) ≤ 2 −n } is an α−open (resp. C−open) subset of X. Now setting H n = {x ∈ X : (f − g)(x) > 2 −n } for every n ∈ N, then by Lemma 3.4, H n is an open subset of X and we have {x ∈ X : (f − g)(x) > 0} = ∪ ∞ n=1 H n and for every n ∈ N, H n and A(f − g, 2 −n ) are disjoint subsets of X. By Lemma 3.2, H n and A(f − g, 2 −n ) can be completely separated by contra-continuous functions. Hence by Theorem 2.2, X has the strong cc−insertion property for (cαc, cc) (resp. (cCc, cc)).