_{1}

^{*}

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

The concept of a C-open set in a topological space was introduced by E. Hatir, T. Noiri and S. Yksel in 1996 [

Recall that a subset A of a topological space

The concept of a set A was β-open if and only if

Recall that a real-valued function f defined on a topological space x was called A-continuous if the preimage of every open subset of

Hence, a real-valued function f defined on a topological space x is called c-continuous (resp. α-continuous) if the preimage of every open subset of

Results of Katĕtov in 1951 [

If g and f are real-valued functions defined on a space X, we write

The following definitions were modifications of conditions considered in paper introduced by E. Lane in 1976 [

A property p defined relative to a real-valued function on a topological space is a c-property provided that any constant function has property p and provided that the sum of a function with property p and any continuous function also has property p. If

In this paper, it is given a sufficient condition for the weak c-insertion property. Also several insertion theorems are obtained as corollaries of this result.

Before giving a sufficient condition for insertability of a continuous function, the necessary definitions and terminology are stated.

Let

Definition 2.1. Let a be a subset of a topological space

Respectively, we have

The following first two definitions are modifications of conditions considered in [

Definition 2.2. If ρ is a binary relation in a set S then

Definition 2.3. A binary relation ρ in the power set

1) If

2) If

3) If

The concept of a lower indefinite cut set for a real-valued function was defined [

Definition 2.4. If f is a real-valued function defined on a space x and if

We now give the following main result:

Theorem 2.1. Let g and f be real-valued functions on a topological space x with

Proof. Let g and f be real-valued functions defined on x such that

Define functions F and g mapping the rational numbers

For any x in x, let

We first verify that

Also, for any rational numbers

The above proof used the technique of proof of Theorem 1 of [

The abbreviations

Corollary 3.1. If for each pair of disjoint α-closed (resp. c-closed) sets

Proof. Let g and f be real-valued functions defined on the X, such that f and g are

since

Corollary 3.2. If for each pair of disjoint α-closed (resp. c-closed) sets

Proof. Let f be a real-valued α-continuous (resp. c-continuous) function defined on the X. Set

Corollary 3.3. If for each pair of disjoint subsets

Proof. Let g and f be real-valued functions defined on the X, such that g is ac (resp.

since

This research was partially supported by Centre of Excellence for Mathematics(University of Isfahan).

Majid Mirmiran, (2016) Weak Insertion of a Continuous Function between Two Comparable α-Continuous (C-Continuous) Functions. Open Access Library Journal,03,1-4. doi: 10.4236/oalib.1102453