Set Star-Menger and Set Strongly Star-Menger Spaces

Definition 1 ([14]).

A space X is said to have the set-Menger property (set-M) if for each nonempty set A ⊂ X and each sequence Un:n∈ℕ of collections of open sets in X such that A¯⊂∪Un, there is a sequence Vn:n∈ℕ such that for each n∈ℕ,Vn is a finite subset of Un and A⊂∪n∈ℕ∪Vn.

Remark 1.

In [19] and [13], the authors noticed that set-Menger spaces are nothing but another way to define Menger spaces.

Definition 2 ([2, 7, 17]).

A space X is said to be:

1. starcompact (SC) if for each open cover U of X there is a finite subset V of U such that X=St(∪V,U).

2. strongly starcompact (SSC) if for each open cover U of X, there is a finite subset F of X such that X=St(F,U).

In a similar way, we define the following.

Definition 3.

A space X is said to be:

1. set-starcompact (set-SC) if for each nonempty subset A of X and each open cover U of A¯, there is a finite subset V of U such that A=St(∪V,U)∩A.

2. set strongly starcompact (set-SSC) if for each nonempty subset A of X and each open cover U of A¯, there is a finite subset F of A¯ such that A=St(F,U)∩A.

   It is clear by the definition that every set strongly starcompact space is set-starcompact.

Theorem 1.1.

Every countably compact space is set strongly starcompact.

Proof. Let X be a countably compact space, A a nonempty subset of X and U a cover of A¯ by open sets in X. As closed subsets of a countably compact space are also countably compact, A¯ is countably compact. But any countably compact space is strongly starcompact [7]. Therefore, there is a finite set F⊂A¯ such that A⊂A¯⊂St(F,U), i.e. X is set strongly starcompact. □

Corollary 1.1.1.

Every countably compact space is set-starcompact.

In [7: Theorem 2.1.5] (see also [9]), it is proved that a strongly starcompact Hausorff space is countably compact. Since set strong starcompactness implies strong starcompactness, we have the following theorem.

Theorem 1.2.

If X is a set strongly starcompact Hausdorff space, then X is countably compact.

Recall that a collection A⊂P(ω) is said to be almost disjoint if each set A∈A is infinite and the sets A ⋂ B are finite for all distinct elements A,B∈A. For an almost disjoint family A, put ψ(A)=A∪ω and topologize ψ(A) as follows: for each element A∈A and each finite set F ⊂ ω, {A} ∪ (A \ F) is a basic open neighborhood of A and the natural numbers are isolated. The spaces of this type are called Isbell-Mrówka ψ-spaces [3,18] or ψ(A) space.

Throughout the paper, notation and terminology mainly follow [8]. The cardinality of a set is denoted by |A|. Let ω denote the first infinite cardinal, ω₁ the first uncountable cardinal, c the cardinality of the set of real numbers. For a cardinal κ, κ⁺ denotes the smallest cardinal greater than κ. For each pair of ordinals α, β with α < β, we write

As usual, a cardinal is an initial ordinal and an ordinal is the set of smaller ordinals. A cardinal is often viewed as a space with the usual order topology.

A subset A of X is said to be regular open if A=Int(A¯). A subset B of X is said to be regular closed if its complement is regular open or, equivalently, if B=Int(B)¯.

Definition 4.

A space X is said to have:

1. the set star-Menger property (shortly, set-SM property) if for each nonempty subset A of X and each sequence Un:n∈ℕ of collections of open sets in X such that A¯⊂∪Un, n ∈ ℕ, there is a sequence Vn:n∈ℕ such that for each n ∈ ℕ, Vn is a finite subset of Un and A⊂∪n∈ℕSt∪Vn,Un;

2. the set strongly star-Menger property (shortly, set-SSM property) if for each nonempty set A ⊂ X and each sequence Un:n∈ℕ of collections of open sets in X such that A¯⊂∪Un, there is a sequence (F_n: n ∈ ℕ) of finite subsets of A¯ such that A⊂∪n∈ℕStFn,Un

   Theorem 1.1 and the following lemma shows that the class of set strongly star-Menger spaces is big enough.

Lemma 2.1.

Every Menger space is set strongly star-Menger.

Proof. Let X be a Menger space. Let A be any nonempty subset of X and Un:n∈ℕ be a sequence of collections of open sets in X such that A¯⊂∪Un. Since closed subsets of Menger spaces are Menger, then A¯ is Menger. Therefore, there exists a sequence Vn:n∈ℕ such that for each n ∈ ℕ, Vn is a finite subset of Un and A¯⊂∪n∈ℕ∪Vn. Choose xV∈A¯∩V for each V∈Vn. For each n ∈ ℕ, let Fn=    xV:V∈Vn. Then each F_n is a finite subset of A¯ and A⊂A¯⊂∪n∈ℕStFn,Un. □

The following example shows that converse of Lemma 2.1 is not true.

Example 1.

There exists a Tychonoff set strongly star-Menger space which is not Menger.

Proof. Let X = [0, ω₁). Then X is a countably compact space, but not Lindelöf. Hence, X is a set strongly star-Menger space which is not Menger. □

We have the following diagram from the definitions and by Lemma 2.1. However, the converse of the implications may not be true as shown by examples below.

DIAGRAM 1

Example 2.

There exists a Tychonoff set star-Menger (resp., set strongly star-Menger) space which is not set-starcompact (resp., set strongly starcompact).

Proof. Let

be the subspace of the product space [0, ω] ×[0, ω]. Then X is a Tychonoff space.

Since X is countable, it is a Menger space. Since every Menger space is set strongly star-Menger and set star-Menger, we have that X is set strongly star-Menger and set star-Menger.

Next, we show that X is not set-starcompact. Let A={⟨n, ω⟩ : n ∈ ω} and for each n ∈ ω, let

Let us consider the open cover

of A=A¯. To prove that X is not set-starcompact, it is enough to show that there exists a point a ∈ A such that a∉St(∪V,U) for any finite subset V of U. So, let V be a finite subset of U. Then, there exists n₁ ∈ ω such that Un∉V for each n > n₁. Let a = 〈m, ω〉 with m > n₁. But U_m is the only element of U containing ⟨m, ω⟩. Thus a=m,ω∉St(∪V,U)=∪V, which shows that X is not set-starcompact. It also follows that X cannot be set strongly starcompact. This completes the proof. □

We are going now to prove in the following two examples that vertical arrows between properties in the second and third lines of Diagram 1 are not reversible.

Example 3.

There exists a Tychonoff star-Menger space which is not set star-Menger.

Proof. Let D(c)=dα:α<c be the discrete space of cardinality c, and let

be the subspace of the product space βD(c)×c++1, where βD(c) is the Stone-Čech compactification of D(c).

In [22: Example 2.2], Song shows that X is a star-Menger space. Now we prove that X is not set star-Menger.

Let A=D(c)×c+. Then A is a closed subset of X. For each α<c, let

Then U_α is open in X by the construction of the topology of X and Uα∩Uα′=0 for α ≠ α′. For each n ∈ ℕ, let Un=Uα:α<c. Then Un:n∈ℕ is a sequence of open covers of A¯. It is enough to show that there exists a point 〈dβ,c+〉∈A such that

for any sequence Vn:n∈ℕ such that Vn is a finite subset of Un,n∈ℕ. Let Vn:n∈ℕ be any sequence such that for each n ∈ ℕ, Vn is a finite subset of Un. For each n ∈ ℕ, Vn is finite, then there exists αn<c such that Uα∉Vn for each α > α_n. Let

If we pick β > α′, then Uβ∉Vn for each n ∈ ℕ. Since U_β is the only element of Un containing the point 〈dβ,c+〉 for each n ∈ ℕ, then 〈dβ,c+〉∉St∪Vn,Un for all n ∈ ℕ, which shows that X is not set star-Menger. □

Example 4.

Assuming ω1<d=c, there exists a Tychonoff pseudocompact strongly star-Menger space which is not set strongly star-Menger.

Proof. Let X=ψ(A) be the Isbell-Mrówka space with A is maximal and |A|=ω1. Then X is strongly star-Menger Tychonoff pseudocompact space (see [3]). Now we prove that X is not set strongly star-Menger. Let A=A=aα:α<ω1. Then A is closed subset of X. For each α < ω₁, let U_α = {a_α} ∪ (a_α). For each n ∈ ℕ, let Un=Uα:α<ω1. Then Un:n∈ℕ is a sequence of open covers of A¯. It is enough to show that there exists a point α_β ∈ A such that

for any sequence (F_n : n ∈ ℕ) of finite subsets of A¯. Let (F_n : n ∈ ℕ) be any sequence of finite subsets of A¯. Then there exists α′ < ω₁ such that Uα∩(∪n∈ℕFn)=0, for each α > α′. Pick β > α′, then Uβ∩Fn=0 for each n ∈ ℕ. Since U_β is the only element of Un containing the point α_β for each n ∈ ℕ, then aβ∉StFn,Un for all n ∈ ℕ. It shows that X is not set strongly star-Menger. □

Example 5.

There exists a T₁ set star-Menger space which is not set strongly star-Menger.

Proof. Let X = A ∪ B, where A=aα:α<c is any set with |A|=c and B is any countable set such that any element of B is not in A. Topologize X as follows: for each α_α ∈ A and each finite subset F ⊂ B, {α_α} ∪ (B \ F) is a basic open neighborhood of a_α, and each element of B is isolated. Then X is a T₁-space.

First we show that X is set star-Menger space. Let C be any nonempty subset of X and Un:n∈ℕ be any sequence of open covers of C¯. We have three possible cases.

1. C ⊂ A.

   Then for each n ∈ ℕ and for each a_α ∈ C, there exists Uα,n∈Un such that a_α ∈ U_α,n, i.e. we can find a finite set F_α,n ⊂ B such that {a_α} ∪ (B \ F_α,n) ⊂ U_α,n. Then, it is clear that for each n ∈ ℕ and each β<c with β ≠ α, Uα,n∩Uβ,n≠0. For each n ∈ ℕ, let Vn=Uα,n. Then each Vn is a finite subset of Un and C⊂∪n∈ℕSt(∪​Vn,Un).

2. C ⊂ B.

   Since B is countable, B is set star Menger. Therefore, for each n ∈ ℕ there is a finite subset Wn of Un such that C⊂∪n∈ℕSt∪Wn,Un.

3. C ⊂ A ∪ B.

   Let C = C₁ ∪ C₂, where C₁ ⊂ A and C₂ ⊂ B. Choose Vn⊂Un and Wn⊂Un such that C1⊂∪n∈ℕSt∪Vn,Un and C2⊂∪n∈ℕSt∪Wn,Un. Let Hn=Vn∪Wn. Then Vn is a finite subset of Hn and C⊂∪n∈ℕSt∪Hn,Un.

So, we conclude that X is a set star-Menger space.

Taking A=A¯⊂X and working quite similarly to the proof of Example 4, we easily prove that X is not a set strongly star-Menger space. □

Problem 1.

Does there exist a Tychonoff set star-Menger space which is not set strongly star-Menger?

Remark 2.

1. The space X in Example 5 is a set-starcompact space but not set strongly starcompact. So, we have proved that the horizontal arrow in the first line of properties in Diagram 1 is not reversible.

2. It is known that there are star-Menger spaces which are not strongly star-Menger (see, for example, [5, 12]) so that the horizontal arrow in the last line of Diagram 1 is not reversible.

In some classes of spaces certain properties from Diagram 1 coincide. Kočinac proved the following theorem in [10].

Theorem 2.1.

([10: Theorem 2.8]). In the class of paracompact Hausdorff spaces, Mengerness, star-Mengerness and strongly star-Mengerness coincide.

From Theorem 2.1 and Diagram 1 we get the following result.

Theorem 2.2.

If X is a paracompact Hausdorff space, then the following statements are equivalent:

1. X is Menger;

2. X is set star-Menger;

3. X is set strongly star-Menger;

4. X is strongly star-Menger;

5. X is star-Menger.

Remark 3.

A space X is said to be para-Lindelöf if every open cover U of X has a locally-countable open refinement V. In [5: Theorem 2.10] it was proved that in the class of regular para-Lindelöf spaces Mengerness, star-Mengerness and strongly star-Mengerness are equivalent. So, Theorem 2.2 can be extended to the class of regular para-Lindelöf spaces.

A space is said to be meta-Lindelöf if every open cover of it has a point-countable open refinement. Kočinac [10] obtained the following corollary.

Corollary 2.2.1.

([10: Corollary 2.10]). A regular meta-Lindelöf strongly star-Menger space is Menger.

Corollary 2.2.1 and Diagram 1 give the following corollary.

Corollary 2.2.2.

If X is a regular meta-Lindelöf space, then the following statements are equivalent:

1. X is Menger;

2. X is set strongly star-Menger;

3. X is strongly star-Menger.

A space X is metacompact if each open cover of X has a point-finite open refinement. From [10: Theorem 2.4], we have the following corollary.

Corollary 2.2.3.

Every set strongly star-Menger hereditarily metacompact spaceXis a Menger space.

Theorem 3.1.

A clopen subspace of a set star-Menger space is also set star-Menger.

Proof. Let X be a set star-Menger space and Y ⊂ X be a clopen subspace. Let A be any subset of Y and Un:n∈ℕ be any sequence of collections of open sets in (Y, τ_Y) such that for each n ∈ ℕ, ClY(A) ⊂ ∪Un. Since Y is open, then Un:n∈ℕ is a sequence of collections of open sets in X, and since Y is closed, Cl_Y(A) = Cl_X(A). Applying the set star-Mengerness of X, there exists a sequence Vn:n∈ℕ such that for each n ∈ ℕ, Vn is a finite subset of Un and A⊂∪n∈ℕ    St∪Vn,Un. Hence Y is set star-Menger. □

On the lines of Theorem 3.1, we can similarly prove the following theorem.

Theorem 3.2.

A clopen subspace of a set strongly star-Menger space is also set strongly star-Menger.

We now consider (non)preservation of set star-Mengerness and set strong star-Mengerness under some sorts of mappings.

Theorem 3.3.

A continuous image of a set star-Menger space is set star-Menger.

Proof. Let X be a set star-Menger space and f: X → Y a continuous mapping from X onto Y. Let B be any subset of Y and Vn:n∈ℕ be a sequence of open covers of B¯. Set A = f^←(B). Since ƒ is continuous, for each n ∈ ℕ, Un=f←(V):V∈Vn is a collection of open sets in X with

As X is set star-Menger, there exists a sequence Un′:n∈ℕ such that for each n ∈ ℕ, Un′ is a finite subset of Un and,

Let Vn′=V:f←(V)∈Un′. Then for each n ∈ ℕ, Vn′ is a finite subset of Vn and

Thus Y is a set star-Menger space. □

Theorem 3.4.

A continuous image of a set strongly star-Menger space is set strongly star-Menger.

Now we give a positive result on preimages of set strongly star-Menger spaces. For this we need a new concept defined as follows. We call a space Xnearly set stongly star-Menger if for each A ⊂ X and each sequence Un:n∈ℕ of open covers of X there is a sequence (F_n : n ∈ ℕ) of finite subsets of X such that A⊂∪n∈ℕStFn,Un.

Theorem 3.5.

Letf : X → Ybe an open and closed, finite-to-one continuous mapping from a spaceXonto a set strongly star-Menger spaceY. ThenXis nearly set strongly star-Menger.

Proof. Let A ⊂ X be any nonempty set and Un:n∈ℕ be a sequence of open covers of X. Then B = ƒ(A) is a subset of Y. Let y∈B¯. Then f^←(y) is a finite subset of X, and thus for each n ∈ ℕ, there is a finite subset Un,y of Un such that f←(y)⊂∪Un,y and U∩f←(y)≠0 for each U∈Un,y. Since ƒ is closed, there exists an open neighborhood V_n,y of y in Y such that f←Vn,y⊂∪U:U∈Un,y. Since ƒ is open, we can assume that

Now, for each n ∈ ℕ, Vn=Vn,y:y∈B¯ is an open cover of B¯. Since Y is set strongly star-Menger, there exist a sequence (F_n : n ∈ ℕ) of finite subsets of B¯ such that

Since ƒ is finite-to-one, the sequence (f^←(F_n) : n ∈ ℕ) is a sequence of finite subsets of X. We prove

Let X ∈ A. Then, there exists n ∈ ℕ and y∈B¯ such that f(x) ∈ V_n,y and Vn,y∩Fn≠0. Since

we can choose U∈Un,y with x ∈ U. Then V_n,y ⊂ f(U). Thus U∩f←(Fn)≠0. Hence x∈Stf←Fn,Un. It follows that

Thus X is nearly set strongly star-Menger. □

Theorem 3.6.

If f : X → Y is an open perfect mapping and Y is a set star-Menger space, then X is nearly set star-Menger.

The product of two set star-Menger spaces need not be set star-Menger. In fact, the following well-known example (see [8, 24]) shows that the product of two countably compact (hence set strongly star-Menger) spaces need not be set star-Menger (hence not set strongly star-Menger). Song [23: Example 2.12] proved that the product X × Y is not star-Menger. Thus X × Y is not set star-Menger. Here we give a rough proof for the sake of completeness.

Example 6.

There exist two countably compact spaces X and Y such that X × Y is not set star-Menger.

Proof. Let Dc be the discrete space of cardinality c. We can define X=∪α<ω1Eα, Y=∪α<ω1Fα, where E_α and F_α are subsets of βDc defined inductively so that the following three conditions are satisfied:

1. Eα∩Fβ=D(c) if α ≠ β;

2. Eα≤c and Fα≤c;

3. every infinite subset of E_α (resp., F_α) has an accumulation point in E_α+1(resp., F_α + 1).

Those sets E_α and F_α are well-defined since every infinite closed set in βDc has cardinality 2c (see [24]). The diagonal {〈d,d〉:d∈D(c)} is a discrete open and closed subset of X × Y of cardinality c so that it is not set star-Menger. Thus X × Y is not set star-Menger, since, by Theorem 3.1, open and closed subsets of set star-Menger spaces are set star-Menger. □

Example 7.

There exist a countably compact (hence, set star-Menger) space X and a Lindelöf space Y such that X × Y is not set star-Menger.

Proof. Let X = [0, ω₁) be equipped with the order topology, and let Y = [0, ω₁] with the following topology. Each point α < ω₁ is isolated, and a set U containing ω₁ is open if and only if Y \ U is countable. Then, X is countably compact and Y is Lindelöf. By [23: Example 2.13], X × Y is not star-Menger. Thus it cannot be set star-Menger. □

Above results suggest the following problem.

Problem 2.

Is the product of a set strongly star-Menger (set star-Menger) space with a compact space set strongly star-Menger (set star-Menger)?

Definition 5.

A space X is said to be:

1. set star-Hurewicz (resp., set strongly star-Hurewicz) if for each A ⊂ X and each sequence Un:n∈ℕ of open covers of A¯ there is a sequence Vn:n∈ℕ (resp., a sequence (F_n : n ∈ ℕ)) such that Vn is a finite subset of Un (resp., F_n is a finite subset of A¯) for each n ∈ ℕ, and each X ∈ A belongs to all but finitely many sets St∪Vn,Un (resp., to all but finitely many sets StFn,Un);

2. set star-Rothberger (resp., set strongly star-Rothberger) if for each A ⊂ X and each sequence Un:n∈ℕ of collections of open sets in X such that A¯⊂∪Un, there is a sequence (U_n : n ∈ ℕ) (resp., a sequence (x_n : n ∈ ℕ)) such that Un∈Un (resp., xn∈A¯) for each n ∈ ℕ and A⊂∪n∈ℕStUn,Un (resp., A⊂∪n∈ℕSt(xn,Un)).

Definition 6.

A space X is said to be:

1. weakly set star-Menger (almost set star-Menger, faintly set star-Menger) if for each A ⊂ X and each sequence Un:n∈ℕ of open covers of A¯ by open sets in X, there is a sequence Vn:n∈ℕ such that Vn is a finite subset of Un for each n ∈ ℕ, and A⊂∪n∈ℕSt∪Vn,Un¯, A⊂∪n∈ℕSt(∪Vn,Un)¯(A⊂∪n∈ℕSt(∪Vn¯,Un));

2. weakly set strongly star-Menger (almost set strongly star-Menger) if for each A ⊂ X and each sequence Un:n∈ℕ of open covers of A¯ there is a sequence (F_n : n ∈ ℕ) of finite subsets of A¯ such that A⊂∪n∈ℕSt(Fn,Un)¯(A⊂∪n∈ℕSt(Fn,Un)¯)..

In a similar way we define weaker forms of Hurewicz-type and Rothberger-type properties.