Insertion of a Contra-α-continuous Function

A necessary and sufficient condition in terms of lower cut sets is given for the insertion of a contra-α-continuous function between two comparable real-valued functions.


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 ( )  [20], while the concept of a locally dense set was introduced by Corson and Michael [4]. A generalized class of closed sets was considered by Maki in [19]. 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 [19].
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 [5,11]. In the recent literature many topologists had focused their research in the direction of investigating different types of generalized continuity.
Dontchev in [6] introduced a new class of mappings called contra-continuity. Jafari and Noiri in [12,13] 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 [1,3,8,9,10,24].
Hence, a real-valued function f defined on a topological space X is called contra-αcontinuous (resp. contra-semi-continuous, contra-precontinuous) if the preimage of every open subset of R is α-closed (resp. semi-closed, preclosed) 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 contra-αcontinuous function between two comparable real-valued functions.
If g and f are real-valued functions defined on a space X, we write for all x in X.
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 cα-property provided that any constant function has property P and provided that the In this paper, it is given a sufficient condition for the weak cα-insertion property.
Also for a space with the weak cα-insertion property, we give a necessary and sufficient condition for the space to have the cα-insertion property. Several insertion theorems are obtained as corollaries of these results.

The Main Result
Before giving a sufficient condition for insertability of a contra-α-continuous function, the necessary definitions and terminology are stated.
be a topological space. Then the family of all α-open, α-closed, semiopen, semi-closed, preopen and preclosed will be denoted by , τ X pC respectively.
, τ X We define the subsets Λ A and V A as follows: We define the subsets is called the α-kernel (resp. prekernel, semi-kernel) of A.
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 ,

Applications
The abbreviations , c cα cpc and csc are used for contra-α-continuous, contraprecontinuous and contra-semi-continuous, respectively.
Before stating the consequences of Theorems 2.1, 2.2, we suppose that X is a topological space whose α-kernel sets are α-open. ).
csc csc resp Proof. Let g and f be real-valued functions defined on the X, such that f and g are cpc (resp. csc), and .
and by Corollary 3.2, since g and f are contra-α-continuous functions hence h is a contra-α-continuous function.
we obtain a decreasing sequence of α-closed subsets of X with the required properties.
is a countable covering of semi-closed (resp. preclosed) subsets of X, we set for . ,