Mathematics Letters
Volume 2, Issue 3, June 2016, Pages: 25-27

Weak Insertion of an α−Continuous Function

Majid Mirmiran

Department of Mathematics, University of Isfahan, Isfahan, Iran

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To cite this article:

Majid Mirmiran. Weak Insertion of an α−Continuous Function. Mathematics Letters. Vol. 2, No. 3, 2016, pp. 25-27. doi: 10.11648/j.ml.20160203.11

Received: August 23, 2016; Accepted: October12, 2016; Published:October 19, 2016


Abstract: A sufficient condition in terms of lower cut sets are given for the weak α−insertion property and the weak insertion of an α−continuous function between two comparable real-valued functions. Also several insertion theorems are obtained as corollaries of this result.

Keywords: Weak Insertion, Strong Binary Relation, Preopen Set, Semi-Open Set, α−Open Set, Lower Cut Set


1. Introduction

The concept of a preopen set in a topological space was introduced by H. H. Corson and E. Michael in 1964 [1].

A subset A of a topological space (X, τ) is called preopen or locally dense or nearly open if A Int(Cl(A)). A set A is called preclosed if its complement is preopen or equivalently if Cl(Int(A)) A.

The term, preopen, was used for the first time by A. S. Mashhour, M. E. Abd El-Monsef and S. N. El-Deeb [2], while the concept of a , locally dense, set was introduced by H. H. Corson and E. Michael [1].

The concept of a semi-open set in a topological space was introduced by N. Levine in 1963 [3]. A subset A of a topological space (X, τ) is called semi-open [3] if A Cl(Int(A)).

A set A is called semi-closed if its complement is semi-open or equivalently if Int(Cl(A)) A.

Recall that a subset A of a topological space (X, τ) is called α−open if A is the dierence of an open and a nowhere dense subset of X.

A set A is called α−closed if its complement is α−open or equivalently if A is union of a closed and a nowhere dense set.

We have a set is α−open if and only if it is semi-open and preopen.

Recall that a real-valued function f defined on a topological space X is called A−continuous [4] if the preimage of every open subset of R belongs to A, where A is a collection of subset 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 [5,6].

Hence, a real-valued function f defined on a topological space X is called precontinuous (resp. semi-continuous or α−continuous) if the preimage of every open subset of R is preopen (resp. semi-open or α−open) subset of X.

Precontinuity was called by V. Ptk nearly continuity [7]. Nearly continuity or precontinuity is known also as almost continuity by T. Husain [8].

Precontinuity was studied for real-valued functions on Euclidean space by Blumberg back in 1922 [9].

Results of M. Katˇetov [10,11] concerning binary relations and the concept of an indefinite lower cut set for a real-valued function, which is due to F. Brooks [12], are used in order to give a sucient condition for the insertion of an α−continuous function between two comparable real-valued functions.

If g and f are real-valued functions defined on a space X, we write g ≤ f in case g(x) ≤ f(x) for all x in X.

The following definitions are modifications of conditions considered in [13].

A property P defined relative to a real-valued function on a topological space is an α−property provided that any constant function has property P and provided that the sum of a function with property P and any α−continuous function also has property P.

If P1 and P2 are α−property, the following terminology is used:

A space X has the weak α−insertion property for (P1,P2) if and only if for any functions g and f on X such that g ≤ f, g has property P1 and f has property P2, then there exists an α−continuous function h such that g ≤ h ≤ f.

2. The Main Result

Before giving a sucient condition for insertability of an α−continuous function, the necessary definitions and terminology are stated.

Let (X, τ) be a topological space, the family of all α−open, α−closed, semi-open, semi-closed, preopen and preclosed will be denoted by αO(X, τ), αC(X, τ), sO(X, τ), sC(X, τ), pO(X, τ) and pC(X, τ), respectively.

Definition 2.1. Let A be a subset of a topological space (X, τ). Respectively, we define the α−closure, α−interior, s-closure, s-interior, p-closure and p-interior of a set A, denoted by αCl(A), αInt(A), sCl(A), sInt(A), pCl(A) and pInt(A) as follows:

αCl(A)= ∩{F: F A, F αC(X, τ )},        (1)

αInt(A)= {O: O A, O αO(X, τ)},         (2)

sCl(A)= ∩{F: F A, F sC(X, τ)},          (3)

sInt(A)= {O: O A, O sO(X, τ)},        (4)

pCl(A)= ∩{F: F A, F pC(X, τ)} and      (5)

pInt(A)= {O: O A, O pO(X, τ)}.         (6)

Respectively, we have αCl(A), sCl(A), pCl(A) are α−closed, semi-closed, preclosed and αInt(A), sInt(A), pInt(A) are α−open, semi-open, preopen. The following first two definitions are modifications of conditions consid­ered in [10,11].

Definition 2.2. If ρ is a binary relation in a set S then ρ¯is defined as follows: x ρ¯y if and only if yρν implies xρν and uρx implies uρy 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:

i).      If Ai ρBj for any i {1,...,m} and for any j {1,...,n}, then there exists a set C in P (X) such that Ai ρC and CρBj for any i {1,...,m}and any j {1,...,n}.

ii).     If A B, then A ρ¯B.

iii).   If AρB, then αCl(A) B and A αInt(B).

The concept of a lower indefinite cut set for a real-valued function was defined by F. Brooks [12] as follows:

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

{xX:f(x)<l}A(f, l){xX:f(x)≤ l}for a real number l,(7)

then A(f, l) is called a lower indefinite cut set in the domain of f at the level l.

We now give the following main result:

Theorem 2.1. Let g and f be real-valued functions on a topological space X with g ≤ f. If there exists a strong binary relation ρ on the power set of X and if 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 t1 <t2 then A(f, t1) ρA(g, t2), then there exists an α−continuous function h defined on X such that g ≤ h ≤ f.

Proof. Let g and f be real-valued functions defined on 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 t1 <t2 then A(f, t1) ρA(g, t2).

Define functions F and G mapping the rational numbers Qinto the power set of X by F (t)= A(f, t) and G(t)= A(g, t). If t1 and t2 are any elements of Q with t1 <t2, then F (t1) ρ¯F (t2), G(t1) ρ¯ G(t2), and F (t1) ρG(t2). By Lemmas 1 and 2 of [11] it follows that there exists a function H mapping Q into the power set of X such that if t1 and t2 are any rational numbers with t1 <t2, then F (t1) ρH(t2), H(t1) ρH(t2) and H(t1) ρG(t2).

For any x in X, let h(x) = inf{t Q: x H(t)}.    (8)

We first verify that g ≤ h ≤ f: If x is in H(t) then x is in G(t/) for any t/ >t; since x is in G(t/)= A(g, t/) implies that g(x) ≤ t/, it follows that g(x) ≤ t. Hence g ≤ h. If x is not in H(t), then x is not in F (t/) for any t/ <t; since x is not in F (t/)= A(f, t/) implies that f(x) >t/, it follows that f(x) ≥ t. Hence h ≤ f.

Also, for any rational numbers t1 and t2 with t1 <t2, we have h−1(t1,t2)= αInt(H(t2)) \ αCl(H(t1)). Hence h−1(t1,t2) is an α−open subset of X, i. e., h is an α−continuous function on X. •

The above proof used the technique of proof of Theorem 1 of [10].

3. Applications

The abbreviations pc and sc are used for precontinuous and semicontinuous, respectively.

Corollary 3.1. If for each pair of disjoint preclosed (resp. semi-closed) sets F1,F2 of X , there exist α−open sets G1 and G2 of X such that F1 G1, F2 G2 and G1 ∩ G2 = then X has the weak α−insertion property for (pc, pc) (resp. (sc, sc)).

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

such that f and g are pc (resp. sc), and g ≤ f.If a binary relation ρ is defined by AρB in case pCl(A) pInt(B) (resp. sCl(A) sInt(B)), then by hypothesis ρ is a strong binary relation in the power set of X. If t1 and t2 are any elements of Q with t1 <t2, then

A(f,t1){xX:f(x)≤ t1}{xX:g(x)<t2}A(g,t2);   (9)

since {x X: f(x) ≤ t1} is a preclosed (resp. semi-closed) set and since {x X: g(x) <t2} is a preopen (resp. semi-open) set, it follows that pCl(A(f, t1)) pInt(A(g, t2)) (resp. sCl(A(f, t1)) sInt(A(g, t2))). Hence t1 <t2 implies that A(f, t1) ρA(g, t2). The proof follows from Theorem 2.1. •

Corollary 3.2. If for each pair of disjoint preclosed (resp. semi-closed) sets F1,F2, there exist α−open sets G1 and G2 such that F1 G1, F2 G2 and G1 ∩ G2 = then every precontinuous (resp. semi-continuous) function is α−continuous.

Proof. Let f be a real-valued precontinuous (resp. semi-continuous) function defined on the X. Set g = f, then by Corollary 3.1, there exists an α−continuous function h such that g = h = f.•

Corollary 3.3. If for each pair of disjoint subsets F1,F2 of X , such that F1 is preclosed and F2 is semi-closed, there exist α−open subsets G1 and G2 of X such that F1 G1, F2 G2 and G1 ∩ G2 = then X have the weak α−insertion property for (pc, sc) and (sc, pc).

Proof. Let g and f be real-valued functions defined on the X, such that g is pc (resp. sc) and f is sc (resp. pc), with g ≤ f.If a binary relation ρ is defined by AρB in case sCl(A) pInt(B) (resp. pCl(A) sInt(B)), then by hypothesis ρ is a strong binary relation in the power set of X. If t1 and t2 are any elements of Q with t1 <t2, then

A(f, t1){xX:f(x)≤ t1}{xX:g(x)<t2}A(g, t2);  (10)

since {x X: f(x) ≤ t1} is a semi-closed (resp. preclosed) set and since {x X: g(x) <t2} is a preopen (resp. semi-open) set, it follows that sCl(A(f, t1)) pInt(A(g, t2)) (resp. pCl(A(f, t1)) sInt(A(g, t2))). Hence t1 <t2 implies that A(f, t1) ρA(g, t2). The proof follows from Theorem 2.1. •

Acknowledgement

This work was supported by University of Isfahan and Centre of Excellence for Mathematics (University of Isfahan).


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