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
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
↑ ↑
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
∗.
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, Q′} where
Q′ 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= (1,3
Example 18. LetRbe the real line with the topologyτ ={∅, R, Q′}whereQ′
is the set of irrational numbers. Then for the natural number setN,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.
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.
is weakly ω∗-continuous, then there exists an ω∗-open subset B of X containing Theorem 32. Let f :X→Y be a function. The following are equivalent:
(1)f is weakly ω∗-continuous,
(7) ⇒ (1) : Let x ∈ X and U any open set of Y containing f(x). Then x ∈
(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, 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) 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.
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 ⊂ ∪n
j=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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Erdal Ekici
Department of Mathematics, Canakkale Onsekiz Mart University, Terzioglu Campus,
17020 Canakkale, Turkey email: [email protected]
Saeid Jafari
Department of Economics, Copenhagen University,
