ON ω∗-CLOSED SETS AND THEIR TOPOLOGY
Erdal Ekici and Saeid Jafari
Abstract. In [2], Arhangel’ski˘ı introduced the notion ofω-closed sets in relation to countable tightness. This paper deals with a class of sets calledω∗-closed sets and its topology which is stronger than the class of ω-closed sets due to Arhangel’ski˘ı.
2000Mathematics Subject Classification: 54C10, 54D10.
1.Introduction
The word tightness was first introduced in [1]. In [2], Arhangel’ski˘ı introduced the notions of ”countable tightness” and ”ω-closedness”. Spaces that are countably tight are essentially those in which the closure operator is determined by countable sets.
A subsetAof a spaceXis calledω-closed [2] ifCl(B)⊂Afor every countable subset B ⊂A. It is shown in [2, 3] that the family of all ω-open subsets of a space forms a topology for it. A topological space X has countable tightness if every ω-closed subset is closed in X [2]. In the survey [6], Goreham discuss the use of sequences and countable sets in general topology. It is proved that every sequential space and every hereditarily separable space has countable tightness. In particular, every countable space has countable tightness. Also, every perfectly regular countable compact space has countable tightness [13]. On the other hand, countable tightness is concerned directly with Moore-Mr´owka problem which is as follows: ”Is every compact Hausdorff space of countable tightness a sequential space?”. The problem was posed in [9]. Other results connected to the Moore-Mr´owka problem can be found in [10]. The purpose of this paper is to introduce and study a class of sets called ω∗-closed sets and its topology which is stronger than the class of ω-closed sets due to Arhangel’ski˘ı.
2.Preliminaries
Throughout this paper, by a space we will always mean a topological space. For a subset Aof a spaceX, the closure and the interior of Awill be denoted by Cl(A) and Int(A), respectively.
A subsetAof a topological spaceXis said to be regular open [11] (resp. regular closed [11], preopen [8]) ifA=Int(Cl(A)) (resp. A=Cl(Int(A)),A⊂Int(Cl(A))).
A point x ∈ X is said to be in the θ-closure [12] of a subset A of X, denoted by θ-Cl(U), if Cl(U)∩A 6= ∅for each open set U of X containing x. A subset A of a space X is called θ-closed if A = θ-Cl(A). The complement of a θ-closed set is called θ-open. The θ-interior of a subset A of X is the union of all open sets of X whose closures are contained in A and is denoted by θ-Int(A). The family of all θ-open subsets of a space (X, τ) is denoted by τθ.
Definition 1. ([2])A subset A of a space (X, τ) is called (1)ω-closed if Cl(B)⊂A for every countable subset B ⊂A.
(2)ω-open if its complement is an ω-closed set.
The family of allω-open subsets of a space (X, τ) is denoted byτω. It was shown that τω is a topology onX [2, 3]. A function f :X→Y is said to be θ-continuous iff−1(V) isθ-open in X for each open subset ofY. A function f :X→Y is called weakly continuous [7] if for eachx∈Xand each open subsetV inY containingf(x), there exists an open subsetU inXcontainingxsuch thatf(U)⊂Cl(V). The graph of a function f :X →Y, denoted byG(f), is the subset {(x, f(x)) :x ∈X} of the product spaceX×Y. For a functionf :X→Y, the graph functiong:X →X×Y of f is defined by g(x) = (x, f(x)) for each x∈X. A subsetA of a space X is said to be N-closed relative toX [4] if for each cover {Ai :i∈I} of A by open sets of X, there exists a finite subfamilyI0⊂I such that A⊂ ∪i∈I0Cl(Ai).
3.ω∗-closed sets and their topology Definition 2. A subsetA of a space (X, τ) is called
(1)ω∗-closed ifθ-Cl(B)⊂A for every countable subset B ⊂A.
(2)ω∗-open if its complement is an ω∗-closed set.
The family of allω∗-open subsets of a space (X, τ) is denoted by τω∗. Remark 3. The following diagram holds for a subset A of a space (X, τ):
ω∗-open −→ ω-open
↑ ↑
θ-open −→ open
None of these implications is reversible as shown in the following examples:
Example 4. Let (R, τco-countable) be the countable complement topological space with the real line. Then the set (0,1) is ω-open but it is not open.
Example 5. (R, τu) be the usual topological space with the real line. Then the set (0,1)is ω-open but it is notω∗-open.
Question: Does there exist a set in a topological space (X, τ) which isω∗-open and it is not θ-open?
Theorem 6. Let (X, τ) be a regular space and A ⊂ X. The following are equivalent:
(1)A is ω-open, (2)A is ω∗-open.
Proof. It follows from the fact thatτ =τθ in a regular space (X, τ).
Theorem 7. Let A be a subset of a space (X, τ). The following are equivalent:
(1)A is ω∗-open,
(2)A⊂θ-Int(X\B) for every countable subset B of X such that A⊂X\B.
Proof. (1)⇒(2) : LetAbeω∗-open andB be a countable subset ofX such that A ⊂ X\B. We have X\A is ω∗-closed. Since B ⊂ X\A and B is countable, then θ-Cl(B)⊂X\A. Hence,A⊂X\θ-Cl(B) =θ-Int(X\B).
(2) ⇒ (1) : Let A ⊂ θ-Int(X\B) for every countable subset B such that A ⊂ X\B. LetB ⊂X\Abe a countable subset. ThenA⊂X\BandA⊂θ-Int(X\B) = X\θ-Cl(B). Thus,θ-Cl(B)⊂X\A and X\A isω∗-closed. Hence,A isω∗-open.
Corollary 8. Let A be a subset of a space(X, τ). The following are equivalent:
(1)A isω∗-open,
(2) A ⊂ θ-Int(B) for every subset B having countable complement such that A⊂B,
(3)A⊂θ-Int(B) for everyB ∈τco-countable such that A⊂B, where τco-countable
is the countable complement topology on X.
Corollary 9. Let A be a subset of a space(X, τ). The following are equivalent:
(1)A isω∗-closed,
(2) θ-Cl(B)⊂A for every B ∈τ∗co-countable such that B ⊂A, where τ∗co-countable
is the family of closed sets of (X, τco-countable).
Theorem 10. For a topological space (X, τ),(X, τω∗) is a topological space.
Proof. It is obvious that ∅, X∈τω∗.
Let A, B ∈ τω∗. This implies that X\A and X\B is ω∗-closed. Let V be a countable subset such that V ⊂X\(A∩B) = (X\A)∪(X\B). There exist the sets
V1 and V2 such that V =V1∪V2 and V1 ⊂X\A and V2 ⊂ X\B. Since X\A and X\B areω∗-closed, thenθ-Cl(V1)⊂X\Aandθ-Cl(V2)⊂X\B. Also,θ-Cl(V) =θ- Cl(V1 ∪V2) = θ-Cl(V1)∪θ-Cl(V2) ⊂ (X\A)∪(X\B). Thus, (X\A)∪(X\B) = X\(A∩B) isω∗-closed and henceA∩B ∈τω∗.
Let{Ai}i∈I be a family ofω∗-open subsets ofX. Then{X\Ai}i∈I is a family of ω∗-closed subsets of X. Suppose thatV ⊂ ∩i∈I(X\Ai) is a countable subset. Then V ⊂ X\Ai for each i ∈I. Since X\Ai is an ω∗-closed subset of X for each i∈ I, then θ-Cl(V) ⊂ X\Ai for each i ∈ I. Thus, θ-Cl(V) ⊂ ∩i∈I(X\Ai) and hence,
∩i∈I(X\Ai) is anω∗-closed subset ofX. Thus, ∪i∈IAi is anω∗-open subset of X.
Example 11. Let (X, τco-countable) be the countable complement topological space. Then (X, τω) is the discrete topological space. It is obvious thatτω ⊂P(X) where P(X) is the power set of X. Let A ⊂ X and C ∈ τco-countable such that A⊂C. Then A⊂Int(C) =C. This implies that A∈τω. Thus,τω is the discrete topology.
Definition 12. Let X be a topological space. The intersection of all ω∗-closed (resp. ω-closed) sets of X containing a subset A is called the ω∗-closure (resp. ω- closure) ofA and is denoted byω∗-Cl(A) (resp. ω-Cl(A)). The union of allω∗-open (resp. ω-open) sets of a space X contained in a subset A is called the ω∗-interior (resp. ω-interior) of A and is denoted by ω∗-Int(A) (resp. ω-Int(A)).
Remark 13. For every open setA, we haveω-Int(A) =Int(A) but the converse is not true as shown in the following example:
Example 14. Let R be the real line with the topology τ = {∅, R, Q0} where Q0 is the set of irrational numbers. Thenω-Int(N) =Int(N) whereN is the set of natural numbers but it is not open.
Theorem 15. Let A be a subset of a spaceX. The following are equivalent:
(1)A is open,
(2) A isω-open and ω-Int(A) =Int(A).
Proof. (1) ⇒(2) : Since every open set is ω-open and ω-Int(A) =Int(A), it is obvious.
(2) ⇒ (1) : Let A be an ω-open and ω-Int(A) = Int(A). Then A = ω- Int(A) =Int(A). Thus,A is open.
Remark 16. If A is an ω∗-open set, then ω∗-int(A) = ω-int(A). If A is a θ-open set, then ω∗-int(A) =θ-int(A). None of these implications is reversible as shown in the following examples.
Example 17. LetR be the real line with the topologyτ ={∅, R,(2,3)}. Then for the set A= (1,32),ω∗-int(A) =θ-int(A) but it is notθ-open.
Example 18. LetRbe the real line with the topologyτ ={∅, R, Q0}whereQ0 is the set of irrational numbers. Then for the natural number set N,ω∗-int(A) =ω- int(A) but it is not ω∗-open.
Theorem 19. Let A be a subset of a spaceX. The following hold:
(1)A is θ-open if and only ifA isω∗-open and ω∗-int(A) =θ-int(A).
(2)A is ω∗-open if and only ifA is ω-open and ω∗-int(A) =ω-int(A).
Proof. (1) : It follows from the fact that everyθ-open isω∗-open andω∗-int(A) = θ-int(A). Conversely, letAbe anω∗-open andω∗-int(A) =θ-int(A). ThenA=ω∗- Int(A) =θ-Int(A). Thus,A isθ-open.
(2) : Since every ω∗-open is ω-open and ω∗-int(A) = ω-int(A), it is obvious.
Conversely, let A be an ω-open and ω∗-int(A) = ω-int(A). Then A= ω-int(A) = ω∗-int(A). Hence,A isω∗-open.
4.The related functions via ω∗-closed sets
In this section, we introduce the related functions via ω∗-closed sets. Moreover, we investigate properties and characterizations of these classes of functions.
Definition 20. A function f :X →Y is said to be ω∗-continuous if for every x∈X and every open subsetV inY containingf(x), there exists anω∗-open subset U in X containing x such that f(U)⊂V.
Theorem 21. For a function f : (X, τ) → (Y, σ), f is ω∗-continuous if and only if f : (X, τω∗)→(Y, σ) is continuous.
Definition 22. A function f :X → Y is said to be ω-continuous if f−1(V) is ω-open in X for each open subset of Y.
Remark 23. The following diagram holds for a functionf :X→Y: ω∗-continuous −→ ω-continuous
↑ ↑
θ-continuous −→ continuous
None of these implications is reversible as shown in the following examples:
Example 24. Let (R, τco-countable) be the countable complement topological space and (R, τu) be the usual topological space with the real line. Then the identity function i: (R, τco-countable)→(R, τu) is ω-continuous but it is not continuous.
Example 25. Let (R, τu) be the usual topological space with the real line. Let Y ={a, b, c}andσ ={Y,∅,{a},{c},{a, c}}. Define a functionf : (R, τu)→(Y, σ)
as follows: f(x) =
a , ifx∈(0,1)
b , ifx /∈(0,1) . Then f is ω-continuous but it is not ω∗- continuous.
Question: Does there exist a function f : (X, τ) → (Y, σ) which is ω∗-continuous and it is not θ-continuous?
Definition 26. A function f : X → Y is said to be weakly ω∗-continuous (resp. weakly ω-continuous) at x ∈ X if for every open subset V in Y containing f(x), there exists an ω∗-open (resp. ω-open) subset U in X containing x such that f(U)⊂Cl(V). f is said to be weaklyω∗-continuous (resp. weakly ω-continuous) if f is weakly ω∗-continuous (resp. weaklyω-continuous) at every x∈X.
Remark 27. The following diagram holds for a functionf :X→Y: weaklyω∗-continuous −→ weaklyω-continuous
↑ ↑
ω∗-continuous −→ ω-continuous
None of these implications is reversible as shown in the following examples:
Example 28. Let (R, τu) be the usual topological space with the real line. Let X ={a, b, c, d}and σ={X,∅,{a},{c},{a, b},{a, c},{a, b, c},{a, c, d}}. Then the function f : (R, τu)→(X, σ) defined by
f(x) =
a , ifx∈(−∞,0]∪[1,∞) b , ifx /∈(−∞,0]∪[1,∞) is weakly ω-continuous but it is notω-continuous.
Example 29. Let (R, τu) be the usual topological space with the real line.
Then the identity functioni: (R, τu)→(R, τu) is weaklyω-continuous but it is not weakly ω∗-continuous.
Question: Does there exist a function f : (X, τ) → (Y, σ) which is weakly ω∗- continuous and it is not ω∗-continuous?
Definition 30. A function f : X → Y is coweakly ω∗-continuous if for every open subset U inY,f−1(F r(U)) isω∗-closed inX, where F r(U) =Cl(U)\Int(U).
Theorem 31. Let f :X→Y be a function. The following are equivalent:
(1)f is ω∗-continuous,
(2)f is weakly ω∗-continuous and coweakly ω∗-continuous.
Proof. (1)⇒(2) : Obvious.
(2) ⇒ (1) : Let f be weakly ω∗-continuous and coweakly ω∗-continuous. Sup- pose that x ∈ X and A is an open subset of Y such that f(x) ∈ A. Since f
is weakly ω∗-continuous, then there exists an ω∗-open subset B of X containing x such that f(B) ⊂ Cl(A). Also, F r(A) = Cl(A)\A and f(x) ∈/ F r(A). Since f is coweakly ω∗-continuous, then x ∈ B\f−1(F r(A)) is ω∗-open in X. For ev- ery y ∈ f(B\f−1(F r(A))), y = f(x∗) for a point x∗ ∈ B\f−1(F r(A)). We have f(x∗) = y ∈ f(B) ⊂ Cl(A) and y /∈ F r(A). Also, f(x∗) = y /∈ F r(A) and f(x∗)∈A. Thus,f(B\f−1(F r(A)))⊂A and hencef isω∗-continuous.
Theorem 32. Let f :X→Y be a function. The following are equivalent:
(1)f is weakly ω∗-continuous,
(2)ω∗-Cl(f−1(Int(Cl(K))))⊂f−1(Cl(K)) for every subset K ⊂Y, (3)ω∗-Cl(f−1(Int(U)))⊂f−1(U) for every regular closed set U ⊂Y, (4)ω∗-Cl(f−1(U))⊂f−1(Cl(U)) for every open setU ⊂Y,
(5)f−1(U)⊂ω∗-Int(f−1(Cl(U))) for every open set U ⊂Y, (6)ω∗-Cl(f−1(U))⊂f−1(Cl(U)) for each preopen set U ⊂Y, (7)f−1(U)⊂ω∗-Int(f−1(Cl(U))) for each preopen set U ⊂Y.
Proof. (1)⇒(2) : LetK ⊂Y and x∈X\f−1(Cl(K)). Then f(x)∈Y\Cl(K).
There exists an open set U containing f(x) such that U ∩K = ∅. Then Cl(U)∩ Int(Cl(K)) =∅. Sincef is weakly ω∗-continuous, then there exists an ω∗-open set V containingxsuch thatf(V)⊂Cl(U). We haveV∩f−1(Int(Cl(K))) =∅. Hence, x∈X\ω∗-Cl(f−1(Int(Cl(K)))) andω∗-Cl(f−1(Int(Cl(K))))⊂f−1(Cl(K)).
(2)⇒(3) : Let U be any regular closed set inY. Hence,ω∗-Cl(f−1(Int(U))) = ω∗-Cl(f−1(Int(Cl(Int(U)))))⊂f−1(Cl(Int(U))) = f−1(U).
(3)⇒ (4) : Let U be an open subset of Y. SinceCl(U) is regular closed in Y, ω∗-Cl(f−1(U))⊂ω∗-Cl(f−1(Int(Cl(U))))⊂f−1(Cl(U)).
(4)⇒(5) : LetU be any open set ofY. SinceY\Cl(U) is open inY, thenX\ω∗- Int(f−1(Cl(U))) = ω∗-Cl(f−1(Y \Cl(U))) ⊂ f−1(Cl(Y \Cl(U))) ⊂ X\f−1(U).
Thus,f−1(U)⊂ω∗-Int(f−1(Cl(U))).
(5) ⇒ (1) : Let x ∈X and U be any open subset of Y containing f(x). Then x∈f−1(U)⊂ω∗-Int(f−1(Cl(U))). TakeV =ω∗-Int(f−1(Cl(U))). Hence,f(V)⊂ Cl(U) andf is weaklyω∗-continuous atx inX.
(1) ⇒ (6) : Let U be any preopen set of Y and x ∈ X\f−1(Cl(U)). There exists an open set S containingf(x) such thatS∩U =∅. We haveCl(S∩U) =∅.
Since U is preopen, then U ∩Cl(S) ⊂ Int(Cl(U))∩Cl(S) ⊂ Cl(Int(Cl(U))∩S)
⊂ Cl(Int(Cl(U)∩S))⊂ Cl(Int(Cl(U ∩S))) ⊂ Cl(U ∩S) = ∅. Since f is weakly ω∗-continuous and S is an open set containing f(x), there exists an ω∗-open set V in X containing x such that f(V) ⊂ Cl(S). We have f(V)∩U =∅ and hence V ∩f−1(U) =∅. Hence,x∈X\ω∗-Cl(f−1(U)) andω∗-Cl(f−1(U))⊂f−1(Cl(U)).
(6) ⇒ (7) : Let U be any preopen set of Y. Since Y\Cl(U) is open in Y, then X\ω∗-Int(f−1(Cl(U))) = ω∗-Cl(f−1(Y \Cl(U))) ⊂ f−1(Cl(Y \Cl(U))) ⊂ X\f−1(U). Hence,f−1(U)⊂ω∗-Int(f−1(Cl(U))).
(7) ⇒ (1) : Let x ∈ X and U any open set of Y containing f(x). Then x ∈ f−1(U) ⊂ ω∗-Int(f−1(Cl(U))). Take V = ω∗-Int(f−1(Cl(U))). Then f(V) ⊂ Cl(U). Thus,f is weakly ω∗-continuous atx inX.
Theorem 33. For a function f :X → Y, f :X →Y be weakly ω∗-continuous at x∈X if and only if x∈ω∗-Int(f−1(Cl(A))) for each neighborhood A of f(x).
Proof. Let A be any neighborhood of f(x). There exists an ω∗-open set B containing x such that f(B) ⊂ Cl(A). Since B ⊂ f−1(Cl(A)) and B is ω∗-open, then x∈B ⊂ω∗-Int(B)⊂ω∗-Int(f−1(Cl(A))).
Conversely, letx∈ω∗-Int(f−1(Cl(A))) for each neighborhoodA of f(x). Take U = ω∗-Int(f−1(Cl(A))). Thus, f(U) ⊂ Cl(A) and U is ω∗-open. Hence, f is weakly ω∗-continuous at x∈X.
Theorem 34. Let f :X→Y be a function. The following are equivalent:
(1)f is weakly ω∗-continuous,
(2)f(ω∗-Cl(G))⊂θ-Cl(f(G)) for each subsetG⊂X, (3)ω∗-Cl(f−1(A))⊂f−1(θ-Cl(A)) for each subsetA⊂Y,
(4)ω∗-Cl(f−1(Int(θ-Cl(A))))⊂f−1(θ-Cl(A))for every subset A⊂Y.
Proof. (1)⇒ (2) : Let G⊂X and x∈ ω∗-Cl(G). Let U be any open set of Y containingf(x). Then there exists anω∗-open setB containingx such thatf(B)⊂ Cl(U). Sincex∈ω∗-Cl(G), thenB∩G6=∅. Thus,∅ 6=f(B)∩f(G)⊂Cl(U)∩f(G) and f(x)∈θ-Cl(f(G)). Hence, f(ω∗-Cl(G))⊂θ-Cl(f(G)).
(2) ⇒ (3) : Let A ⊂ Y. Then f(ω∗-Cl(f−1(A))) ⊂ θ-Cl(A). Hence, ω∗- Cl(f−1(A))⊂f−1(θ-Cl(A)).
(3) ⇒ (4) : Let A ⊂ Y. Since θ-Cl(A) is closed in Y, then ω∗-Cl(f−1(Int(θ- Cl(A)))) ⊂f−1(θ-Cl(Int(θ-Cl(A))))) =f−1(Cl(Int(θ-Cl(A))))) ⊂f−1(θ-Cl(A)).
(4)⇒(1) : Let U be any open set of Y. Then U ⊂Int(Cl(U)) =Int(θ-Cl(U)).
Hence,ω∗-Cl(f−1(U))⊂ω∗-Cl(f−1(Int(θ-Cl(U))))⊂f−1(θ-Cl(U)) =f−1(Cl(U)).
By Theorem 32, f is weaklyω∗-continuous.
Definition 35. If a space X can not be written as the union of two nonempty disjoint ω∗-open sets, then X is said to beω∗-connected.
Theorem 36. If f : X → Y is a weakly ω∗-continuous surjection and X is ω∗-connected, then Y is connected.
Proof. Suppose thatY is not connected. There exist nonempty open setsAand B of Y such that Y = A∪B and A∩B = ∅. Then A and B are clopen in Y. By Theorem 32, f−1(A) ⊂ ω∗-Int(f−1(Cl(A))) = ω∗-Int(f−1(A)). Hence f−1(A) is ω∗-open inX. Similarly, f−1(B) is ω∗-open inX. Hence, f−1(A)∩f−1(B) =∅, X = f−1(A)∪f−1(B) and f−1(A) and f−1(B) are nonempty. Thus, X is not ω∗-connected.
Theorem 37. Let {Ai : i ∈ I} be an ω∗-open cover of a space X. Then a function f : X → Y is weakly ω∗-continuous if and only if for each i ∈ I, the restriction fAi :Ai →Y is weakly ω∗-continuous.
Proof. Obvious.
Theorem 38. Let f : X → Y be weakly ω∗-continuous and Y be Hausdorff.
The following hold:
(1) for each (x, y) ∈/ G(f), there exist an ω∗-open set G ⊂ X and an open set U ⊂Y containing x and y, respectively, such that f(G)∩int(cl(U)) =∅.
(2)inverse image of each N-closed set of Y is ω∗-closed inX.
Proof. (1) : Let (x, y)∈/G(f). Theny6=f(x). SinceY is Hausdorff, there exist disjoint open sets U and V containing y and f(x), respectively. Thus,int(cl(U))∩ cl(V) =∅. Sincefis weaklyω∗-continuous, there exists anω∗-open setGcontaining x such that f(G)⊂cl(V). Hence,f(G)∩int(cl(U)) =∅.
(2) : Suppose that there exists aN-closed set W ⊂Y such that f−1(W) is not ω∗-closed in X. There exists a point x ∈ ω∗-cl(f−1(W))\f−1(W). Since f(x) ∈/ f−1(W), then (x, y)∈/ G(f) for eachy∈W. Then there existω∗-open setsGy(x)⊂ X and an open setB(y)⊂Y containingxandy, respectively, such thatf(Gy(x))∩ int(cl(B(y))) =∅. The family {B(y) :y ∈W} is a cover of W by open sets of Y. Since W is N-closed, there exit a finite number of points y1, y2, ..., yn inW such that W ⊂ ∪nj=1int(cl(B(yj))). Take G=∩nj=1Gyj(x). Then f(G)∩W =∅. Since x∈ω∗-cl(f−1(W)), thenf(G)∩W 6=∅. This is a contradiction.
Theorem 39. For a function f : X → Y, f is weakly ω∗-continuous if the graph function g is weakly ω∗-continuous.
Proof. Let g be weakly ω∗-continuous and x ∈ X and A be an open set of X containing f(x). Then X×A is an open set containing g(x). Then there exists an ω∗-open setB containing x such that g(B) ⊂Cl(X×A) =X×Cl(A). Thus, f(B)⊂Cl(A) andf is weaklyω∗-continuous.
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[11] M. H. Stone,Applications of the theory of Boolean rings to general topology, TAMS, 41 (1937), 375-381.
[12] N. V. Velicko, H-closed topological spaces, Amer. Math. Soc, Transl., 78 (1968). 102-118.
[13] W. Weiss,Countable compact spaces and Martin’s Axioms, Canadian Journal of Mathematics, 30 (1978), 243-249.
Erdal Ekici
Department of Mathematics, Canakkale Onsekiz Mart University, Terzioglu Campus,
17020 Canakkale, Turkey email: [email protected] Saeid Jafari
Department of Economics, Copenhagen University,
Oester Farimagsgade 5, building 26, 1353 Copenhagen K, Denmark email: [email protected]
