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 diﬀerence 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 suﬃcient 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 suﬃcient 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
{x∈X:f(x)<l}⊆A(f, l)⊆{x∈X: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)⊆{x∈X:f(x)≤ t1}⊆{x∈X: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)⊆{x∈X:f(x)≤ t1}⊆{x∈X: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).
References