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Set-Menger and related properties
Journal Pre-proof Set-Menger and related properties
Ljubiša D.R. Koˇcinac, Sükran ¸ Konca
PII:
S0166-8641(19)30401-8
DOI:
https://doi.org/10.1016/j.topol.2019.106996
Reference:
TOPOL 106996
To appear in:
Topology and its Applications
Received date:
24 December 2018
Revised date:
21 February 2019
Accepted date:
22 February 2019
Please cite this article as: L.D.R. Koˇcinac, S. ¸ Konca, Set-Menger and related properties, Topol. Appl. (2019), 106996, doi: https://doi.org/10.1016/j.topol.2019.106996.
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Set-Menger and related properties Ljubiˇ sa D.R. Koˇ cinac University of Niˇs, Faculty of Sciences and Mathematics, 18000 Niˇs, Serbia [email protected] S ¸u ¨ kran Konca Department of Mathematics, Bitlis Eren University, 13000 Bitlis, Turkey [email protected] Abstract In this work, we consider new types of covering properties called setMenger, set-Rothberger, set-Hurewicz and set-Gerlits-Nagy. We investigate the relationships between set-Mengerness and some other Mengertype covering properties (such as Mengerness, weak Mengerness, quasiMengerness) and study topological properties of set-Menger and related spaces. In particular, we give a game-theoretic characterization of setMenger spaces.
2010 Mathematics Subject Classification: Primary 54D20, Secondary 54B05, 54C10, 91A44 Keywords: Selection principles, game theory, Menger, set-Menger, quasi-Menger
1
Introduction
Recently, a number of papers on weaker forms of classical covering properties of Menger, Hurewicz and Rothberger has been published [3, 4, 6, 10, 11, 15, 17, 22]. In the last few years the interest to the study of weaker forms of these classical selection properties increased. Di Maio and Koˇcinac [6] have defined quasiMenger and quasi-Rothberger spaces and proved that they are different from other weak versions of Menger’s and Rothberger’s properties. In [2], Arhangel’skii defined a cardinal function sL, and spaces X such that sL(X) = ω we call sLindel¨ of : a space X is sLindel¨ of if for each subset A of X and each cover U of A by sets open in X there is a countable set V ⊂ U such that A ⊂ ∪V. Following this idea and modifying it we here consider new types of selective covering properties which will be called set covering properties. We mainly 1
investigate set-Menger and set-Rothberger properties and their relationships with other Menger-type and Rothberger-type covering properties. Throughout the paper by “a space” we mean “a topological space”. By N, R, Q and P we denote the set of natural, real, rational and irrational numbers, respectively. We use usual topological notation and terminology as in [7]. If P is a collection of subsets of a space X, we write P := {P : P ∈ P}, P := {P : P ∈ P}. If f : X → Y is a mapping between spaces X and Y , and B ⊂ Y we prefer to use f ← (B) for the preimage of B instead of f −1 (B).
2
Known definitions
Recall some definitions concerning selection principles theory and properties which will be considered in this article (see [12, 18]). A space X is said to have: 1. the Menger property M [14] (respectively, Rothberger property R [16]) if for each sequence (Un )n∈N of open covers of X there exists a sequence (Vn )n∈N (respectively, a sequence (Un )n∈N ) such that for eachn ∈ N, Vn is a finite subset of Un (respectively, Un ∈ Un , n ∈ N) and n∈N Vn = X (respectively, n∈N Un = X); 2. the Hurewicz property H if for each sequence (Un )n∈N of open covers there Vn is a finite subset of Un and each is a sequence (Vn )n∈N such that for each n, x ∈ X belongs to all but finitely many sets Vn [9]; 3. the weak Menger property wM [5] (respectively, almost Menger property asequence aM [11]) if for each sequence (Un )n∈N of open covers of X there exists (Vn )n∈N such that for each n ∈ N, Vn is a finite subset of Un and n∈N Vn = X (respectively, n∈N {V : V ∈ Vn } = X); 4. the weak Rothberger property wR [5] (respectively, almost Rothberger property aR [11]) if for each sequence (Un )n∈N of open covers of X there exists a sequence (Un )n∈N such that Un ∈ Un , n ∈ N, and n∈N Un = X (respectively, n∈N Un = X); 5. the almost Hurewicz property aH [22] (respectively, almost Gerlits-Nagy property aGN) if for each sequence (Un )n∈N of open covers of X there is a sequence (Vn )n∈N (respectively, a sequence (Un )n∈N ) such that for each n, Vn is a finite subset of Un (respectively, Un ∈ Un ) and each x ∈ X belongs to Vn (respectively, Un ) for all but finitely many n; 6. the quasi-Menger property qM [6] (respectively, quasi-Rothberger property qR [6]) if for each closed set F ⊂ X and each sequence (Un )n∈N of covers of F by sets open in X there is a sequence (Vn )n∈N (respectively, a sequence (Un )n∈N ) such that finite subset of Un (respectively, Un ∈ Un ) and for each n, Vn is a Vn ⊃ F (respectively, n∈N Un ⊃ F ); n∈N 7. the quasi-Hurewicz property qH [6] (respectively, quasi-Gerlits-Nagy property qGN [6]) if for each closed set F ⊂ X and each sequence (Un )n∈N of covers of F by sets open in X there is a sequence (Vn )n∈N (respectively, a sequence (Un )n∈N ) such that for each n ∈ N, Vnis a finite subset of Un (respectively, Un ∈ Un ) and each x ∈ F belongs to Vn (respectively, to Un ) for all but finitely many n. 2
3
New definitions
Definition 3.1 Let P be a family of nonempty subsets of a space X. We say that X is: (1) P-Menger (respectively, weakly P-Menger, almost P-Menger ) if for each A ∈ P and each sequence (Un )n∈N of sets open in X such that A ⊂ ∪Un for each n ∈ N, there is a sequence (Vn )n∈N , Vn is a finite subset of Un for each n ∈ N and A ⊂ n∈N Vn (respectively, A ⊂ n∈N Vn , A ⊂ n∈N Vn ); (2) P-Rothberger (respectively, weakly P-Rothberger, almost P-Rothberger ) if for each A ∈ P and each sequence (Un )n∈N of sets open in X such that A ⊂ Un , n ∈ N, there is a sequence (Un )n∈N such that Un ∈ Un for each n ∈ N and A ⊂ n∈N Un (respectively, A ⊂ n∈N Un , A ⊂ n∈N Un ); (3) P-Hurewicz (respectively, almost P-Hurewicz ) if for each A ∈ P and each sequence (Un )n∈N of sets open in X such that A ⊂ Un , n ∈ N, there is Un for each n ∈ N a sequence (Vn )n∈N such that Vn is a finite subset of and each x ∈ A belongs to all but finitely many sets Vn (respectively, to all but finitely many sets Vn ); (4) P-Gerlits-Nagy (respectively, almost P-Gerlits-Nagy) if for each A ∈ P and each sequence (Un )n∈N of sets open in X such that A ⊂ ∪Un , there is a sequence (Un )n∈N such that Un ∈ Un for each n ∈ N and each x ∈ A belongs to all but finitely many sets Un (respectively, all but finitely many sets Un ). To give some examples we need several definitions. The depth of a space X, denoted by g(X), is the supremum of cardinalities of closures of discrete subspaces of X [1]. Let κ be an infinite cardinal, p a filter on κ, and (xα )α∈κ a κ-sequence in a space X. A point x ∈ X is called a p-limit point of (xα )α∈κ , written plim xα = x, if for any neighbourhood U of x, {β ∈ κ : xβ ∈ U } ∈ p. Denote by μκ the set of all uniform ultrafilters on κ (p ∈ βκ \ κ is in μκ if |p| = κ). For p ∈ μκ, a space X is p-κ-compact if each κ-sequence (xα )α∈κ in X has a p-limit point in X. Let κ be an infinite cardinal number. A space X is said to be: • κ-bounded if the closure of any A ⊂ X of cardinality ≤ κ is compact (see, for example, [19, 20]); • κ-ultracompact if λ < κ and p ∈ μλ imply X is p-λ-compact [20]. Call a space X κ-σ-bounded if the closure of any A ⊂ X of cardinality ≤ κ is σ-compact.
3
Example 3.2 (1) Let P≤κ be the set of all nonempty subsets of X having cardinality ≤ κ. Then X is P≤κ -Menger provided X is κ-σ-bounded (in particular, κ-bounded). (2) Every κ-ultracompact regular space is P≤κ -Menger. (Recall that by [20, Theorem 5.4], every regular κ-ultracompact space X is κ-bounded.) (3) Let Dc be the set of all discrete closed subspaces of X. If X is g(X)bounded (or g(x)-σ-bounded), it is Dc -Menger. (4) Let L be the set of all Lindel¨of subspaces of a space X in which the closure of any Lindel¨ of subspace is σ-compact. Then X is L-Menger (in fact, L-Hurewicz). If P is the family of all nonempty subsets of X, then we say that X is setMenger (set-M) (set-Rothberger (set-R), set-Hurewicz (set-H), set-Gerlits-Nagy (set-GN)) instead of P-Menger (P-Rothberger, P-Hurewicz, P-Gerlits-Nagy). Similarly for other classes of spaces defined above. We will be concentrated on investigation of set-Menger and set-Rothberger spaces and their relatives.
4
Set-Menger and set-Rothberger spaces
We begin this section with a simple fact. Proposition 4.1 Set-Mengerness implies Mengerness, and almost set-Mengerness implies weak set-Mengerness. Remark 4.2 If X is a Menger space and P ∗ is the set of all (proper) non-dense subsets of X, then X is P ∗ -Menger. Let A be a proper non-dense subset of X and let (Un )n∈N be a sequence of open subsets of X such that A ⊂ Un for each n. For each n ∈ N, the set Vn := Un ∪{X \A} is an open coverof X.Since X is Menger, there are Wn ⊂ Vn for each n, Wn is finite and X = n∈N Wn . The sets Wn = Wn \ {X \ A}, n ∈ N, also cover A and such Wn s witness for (Un )n∈N that X is P ∗ -Menger. Similar assertions are true for set-R, set-H, set-GN properties. Theorem 4.3 Let X be a regular space. If X is almost set-Menger (almost set-Rothberger), then X is set-Menger (set-Rothberger). Proof . We will prove only the Menger case. Let A ⊂X and let (Un )n∈N be a sequence of families of open sets in X such that A ⊂ Un for each n. Since X is a regular space, for each n there exists an open cover Wn of A such that Wn := {W : W ∈ Wn } refines Un and covers A. As X is almost set-Menger, there exists (Hn )n∈N such that for each n, Hn is a finite subset of a sequence Wn and n∈N {H : H ∈ Hn } ⊃ A. For each n ∈ N, and each H ∈ Hn pick VH ∈ Un such that H ⊂ VH . Let Vn := {VH : H ∈ Hn }. Then each Vn is a finite subset of Un and A ⊂ n∈N Vn which shows that X is a set-Menger space. 4
A topological space X is a P -space if the union of countably many closed subsets of X is closed in X. Theorem 4.4 If a P -space X is weakly set-Menger, then X is almost setMenger. Proof . Let A be a subset of X and (Un )n∈N be a sequence of open covers of A by sets open in X. Since X is weakly set-Menger, there exists a sequence of (W n )n∈N such that for each n ∈ N, Wn is a finite subset Un and A ⊂ Wn . As X is a P -space, we have that the set n∈N Wn is closed. n∈N Therefore we have n∈N Wn ⊂ n∈N Wn (because theset on the left side of this inclusion is the smallest closed set containing n∈N Wn ). It follows Wn ⊂ {W : W ∈ Wn }, A⊂ n∈N
n∈N
which means that X is almost set-Menger. Corollary 4.5 Let X be a regular P -space. Then the following statements are equivalent: 1. X is set-Menger; 2. X is almost set-Menger; 3. X is weakly set-Menger. Example 4.6 There is a regular, weakly Rothberger space which is not weakly set-Rothberger. Consider the real line R equipped with the rational sequence topology τ (see [23, Example 65]): for each r ∈ Q the set {r} is open; for each irrational number x let (local) basic sets at x are of the form Uq (x) = {x} ∪ {qi : i ∈ N}, where q = (qk )k∈N is a sequence of rational numbers converging to x in the Euclidean topology. Observe that all basic sets are also closed. This space is regular [23] and weakly Rothberger as it is separable. However, this space is not weakly set-Rothberger. For, consider the subset P ⊂ R of irrational numbers . It is closed and discrete in (R, τ ). For x ∈ P and n ∈ N denote by sn (x) a sequence of rational numbers in (x − 1/2n , x) converging to x and Un (x) = sn (x) ∪ {x}. Let Un = {Un (x) : x ∈ P}. We have the sequence (Un )n∈N of covers of P = P by basic sets open in (R, τ ). If we take one Un ∈ Un , n ∈ N, then clearly P is not covered by n∈N Un . On the other hand, (R, τ ) is not Rothberger (because it is not Lindel¨of [23]), almost set-Rothberger and set-Rothberger. Theorem 4.7 Set-Mengerness implies quasi-Mengerness. Proof . Let F be a closed subset of a set-Menger space X and (Un )n∈N be a sequence of covers of F by sets open in X. Since X is set-Menger, there is finite subset of Un for each n ∈ N, and a sequence (Vn )n∈N such that Vnis a F ⊂ n∈N Vn . Then F = F ⊂ n∈N Vn , i.e. X is quasi-Menger. 5
Theorem 4.8 Quasi-Mengerness implies weak set-Mengerness. Proof . Let A be a subset of a quasi-Mengerspace X and let (Un )n∈N be a sequence of open subsets of X such that A ⊂ Un for each n. Apply the fact that X is quasi-Menger to the closed set A and the sequence (Un )n∈N . There is a sequence (Vn )n∈N such thatVn isa finite subset of Un for each n, and A ⊂ n∈N Vn . Therefore, A ⊂ n∈N Vn , i.e. X is weakly-set-Menger. In a similar way we prove the corresponding theorems for other considered classes of spaces. Therefore, we have the following diagram. set−R ⇑ set−GN ⇓ set−H ⇓ set−M ⇑ set−R
⇒ ⇒ ⇒ ⇒ ⇒
R ⇑ GN ⇓ H ⇓ M ⇑ R
⇒
qR ⇒ ⇑ ⇒ qGN ⇒ ⇓ ⇒ qH ⇒ ⇓ ⇒ qM ⇒ ⇑ ⇒ qR ⇒
w−set−R ⇓
⇐
⇓ ⇓ w−set−M ⇐ ⇑ w−set−R ⇐
a−set−R ⇑ a−set−GN ⇓ a−set−H ⇓ a−set−M ⇑ a−set−R
Diagram 1 Example 4.9 There are quasi-Rothberger (so weakly set-Rothberger) spaces which are not set-Rothberger. (1) Let RS be the Sorgenfrey line. By Proposition 2.2 in [6] it is known that RS is quasi-Rothberger, hence weakly set-Rothberger. To show that it is not set-Rothberger, let A = (0, 2) ⊂ RS . Let Un := basic sets in RS having length less than 21n and covering A = [0, 2) . If we take any Un ∈ Un , n ∈ N, then the length of all chosen elements is less than 1 1 1 + + . . . + n + . . . = 1. 2 22 2 But the length of A is 2, so A cannot be covered by selected sets Un , n ∈ N. (2) It is known that a compact space is Rothberger if and only if it is scattered. So, any hereditarily separable compact non-scattered space is weakly set-Rothberger (being quasi-Rothberger), but not set-Rothberger because it is not Rothberger. For example, the closed unit interval I = [0, 1], the Cantor set 2ω and the Hilbert cube I ω are not set-Rothberger, but they are weakly set-Rothberger. (3) According to [17, Theorem 3.4] every hereditarily Lindel¨ of space with ortho-base is hereditarily weakly Rothberger and so weakly set-Rothberger. In particular, every ccc GO space is weakly set-Rothberger; for example, the Souslin line. 6
Example 4.10 The deleted radius topology in the plane [23, Example 77] is not almost Rothberger [3], so it is not almost set-Rothberger. But it is weakly set-Menger. Example 4.11 Set-Mengerness is not a hereditary property. Let R be the real line with the usual metric topology. Assume that A isany subset of R and (Un )n∈N is a sequence of sets open in R satisfying A ⊂ Un for each n ∈ N. Since A is a Menger space as a closed subspace of the Menger space R, there are finite sets Vn ⊂ Un , n ∈ N, such that A ⊂ A ⊂ n∈N Vn . It follows that R is a set-Menger space. Consider the space P of irrational real numbers. It is known that P is not Menger, so it cannot be set-Menger. Remark 4.12 The space P is an example of a weakly set-Menger space (because according to [6, Proposition 2.2] it is quasi-Menger) which is neither setMenger nor almost set-Menger (because it is not almost Menger). Theorem 4.13 If A is a clopen (closed and open) subset of a set-Menger (setRothberger) space X, then A is also set-Menger (set-Rothberger). Proof . Let X be a set-Rothberger space and let B be a subset of (A, τA ) and (Un )n∈N be a sequence of covers of ClτA (B) (the closure of B with respect to the topology τA ) by sets open in (A, τA ). Since A is open, each Un is a family of sets open in (X, τ ). Since A is closed, ClτA (B) = Clτ (B). B is a subset of X, and (Un )n∈N is a sequence of covers of Clτ (B) by sets open inX. Because X is set-Rothberger, there exist Un ∈ Un , n ∈ N, such that B ⊂ n∈N Un . As each Un is open in (A, τA ), it follows that A is set-Rothberger. Quite similarly we prove the set-Menger part of the theorem. Theorem 4.14 Continuous image of a set-Menger (set-Rothberger) space is also set-Menger (set-Rothberger). Proof . We consider only the set-Rothberger case. Assume that X is a setRothberger space and f : X → Y is a continuous mapping from X onto Y . Let B be any subset of Y and (Vn )n∈N be a sequence of covers of B by sets open in Y . Let A = f ← (B). Since f is continuous, for each n ∈ N, f ← (Vn ) = Un is the collection of open sets in X with Vn = Un . A = f ← (B) ⊂ f ← (B) ⊂ f ← ← As X is set-Rothberger, there are Un = f (Vn ) ∈ Un , n ∈ N, such that A ⊂ n∈N Un . So f (Un ) = Vn B = f (A) ⊂ n∈N
n∈N
and we conclude that Y is set-Rothberger. In a similar way, with a small changes in the proof, we get: 7
Theorem 4.15 A continuous image of a weakly (almost) set-Menger space is also weakly (almost) set-Menger. Recall that a mapping f : X → Y is called strongly θ-continuous [13] if for each x ∈ X and each open set V containing f (x) there is an open set U containing x such that f (Cl(U )) ⊂ V . Every strongly θ-continuous mapping is continuous. Theorem 4.16 A strongly θ-continuous image Y = f (X) of an almost setMenger space X is a set-Menger space. Proof . Let B be any subset of Y and let (Vn )n∈N be a sequence of covers of B by sets open in Y . Let A = f ← (B) and let x ∈ A. Since strongly θcontinuous mappings are continuous, we have f (x) ∈ f (f ← (B)) ⊂ f (f ← (B)) = B. Thus for each n ∈ N there is a set V (x, n) ∈ Vn containing f (x). Since f is strongly θ-continuous, there is an open set U (x, n) ⊂ X containing x such that f (U (x, n)) ⊂ V (x, n). Therefore, for each n, the set Un = {U (x, n) : x ∈ A} is an open cover of A. As X is almost set-Menger, there is a sequence n )n∈N (F of finite sets such that for each n, Fn is a subset of Un and A ⊂ n∈N {F : F ∈ Fn }. To each F ∈ Fn assign a set WF ∈ Vn with f (F ) ⊂ WF and put Wn := {WF : F ∈ Fn }. We obtain the sequence (Wn )n∈N of finite subsets of Vn , n ∈ N, which witnesses for (Vn )n∈N that Y is a set-Menger space. What about pre-images of set-Menger spaces? We need a new concept called nearly set-Menger space. Recall that a subset Y of a space X is said to relatively )n∈N of open covers of X, there exist finite Menger in X if for each sequence (Un sets Vn ⊂ Un , n ∈ N, such that n∈N Vn ⊃ Y . A space X is nearly set-Menger if each subset A of X is relatively Menger in X. Theorem 4.17 The perfect preimage of a set-Menger space is nearly set-Menger. Proof . Let f : X → Y be a perfect mapping from a space X onto a setMenger space Y . Let (Un )n∈N be a sequence of open covers of X and A ⊂ X. Then B = f (A) is a subset of Y . Let y ∈ f (A). Then f ← (y) is compact in X so that for each n ∈ N there exists a finite subset Vn,y of Un such that f ← (y) ⊂ ∪Vn,y = Vn,y . Because f is closed, there is an open set Wn,y ⊂ Y such that y ∈ Wn,y and f ← (y) ⊂ f ← (Wn,y ) ⊂ Vn,y . For each n ∈ N the set Wn := {Wn,y : y ∈ f (A)} is a cover of f (A) by sets open in Y . Since Yis set-Menger, for each n there is a finite Hn ⊂ Wn such that f (A) ⊂ n∈N Hn . For each H ∈ Hn there is the corresponding finite Vn,H ⊂ Un . The set Gn := {V : V ∈ Vn,H , H ∈ Hn } is a finite subset of Un . Further, A ⊂ f ← (f (A)) ⊂ Gn . n∈N
This completes the proof.
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Corollary 4.18 The product X × Y of a set-Menger space X and a compact Hausdorff space Y is a nearly set-Menger space. The corollary follows from the fact that the projection mapping X × Y → X is perfect, because Y is compact. Theorem 4.19 If f : X → Y is an open, perfect mapping from a space X onto a weakly set-Menger space Y , then for each set A ⊂ X and each sequence (Un )n∈N of open covers of f ← (f (A)) by sets open in X there is a sequence (Vn )n∈N of finite sets such that for each n, Vn ⊂ Un and n∈N Vn ⊃ f ← (f (A)) ⊃ A. Proof . Let A be a subset of X and (Un )n∈N a sequence of covers of f ← (f (A)) = (because f is open and continuous) by sets open in X. For each y ∈ f (A) the set Cy := f ← (y) is compact so that for each n ∈ N there is Vn (y). As f is a finite set Vn (y) ⊂ Un which covers Cy . Let Vn (y) = a closed mapping, for each n ∈ N and each y ∈ f (A) there is an open set Wn (y) ⊂ Y such that y ∈ Wn (y) and f ← (Wn (y)) ⊂ Vn (y). For each n ∈ N set Wn = {Wn (y) : y ∈ f (A)}. Then each Wn is a cover of f (A) by sets open in Y . Since Y is weakly set-Menger, there is a sequence (Hn )n∈N such that Hn is a finite subset of Wn , n ∈ N, and f (A) ⊂ n∈N Hn . For each UH . If n and each H ∈ Hn there is a finite UH ⊂ Un with f ← (H) ⊂ Gn = {U ∈ Un : U ∈ UH , H ∈ Hn }, then Gn is a finite subset of Un for each n. Since f is open, we have Hn ) = f ← ( Hn ) ⊂ Gn f ← ( Hn ) ⊂ A ⊂ f ← (f (A)) ⊂ f ← ( f ← (f (A))
n∈N
n∈N
n∈N
n∈N
which completes the proof. We end the paper by game-theoretic characterizations of set-Menger-type covering properties. Because the classical Menger property has a game-theoretic characterization it is natural to ask: Problem 4.20 Do set-Menger properties have game-theoretic characterizations? The following game is associated to the set-Menger (respectively, weakly setMenger, almost set-Menger) property. Let A be a fixed subset of a space X. In the game GA (respectively, wGA , aGA ) two players, I and II, play a round for each n ∈ N. In then − th round player I chooses a collection Un of open sets in X such that A ⊂ Un , and II responds by choosing a finite Vn subset of Un . II wins a play U1 , V1 ; U2 , V2 ; . . . , Un , Vn , . . . if A ⊂ n∈N Vn (respectively, A ⊂ n∈N Vn , A ⊂ n∈N Vn ); otherwise I wins. Notice that if I does not have a winning strategy in the game GA (respectively, wGA , aGA ) for each A ⊂ X, then X is set-Menger (respectively, weakly setMenger, almost set-Menger). The next theorems show that the converse is also true. 9
Theorem 4.21 A Lindel¨ of space X is set-Menger if and only if the player I does not have a winning strategy in the game GA for each A ⊂ X. Proof . Let A be a subset of a Lindel¨ of set-Menger space X. Let σ be a strategy that the first move of I is a set σ(∅) = U1 of sets for I in the game GA . Assume open in X such that A ⊂ U1 . One may suppose that U1 = {U(1) ⊂ U(2) ⊂ . . .}. Let for each n ∈ N, σ(U(n) ) be the set {U(n,1) ⊂ U(n,2) ⊂ U(n,3) ⊂ . . .}. We continue in this way and for each m-sequence (n1 , . . . , nm ) of natural numbers we define σ(U(n1 ) , U(n1 ,n2 ) , . . . , U(n1 ,n2 ,...,nm ) ) = {U(n1 ,...,nm ,1) ⊂ U(n1 ,...,nm ,2) ⊂ . . .}. This procedure led to the family {Us : s is a finite sequence in N } such that {U(s,1) , U(s,2) , . . .} is a cover of A by sets open in X. We define now covers Un = {Un,m : m ∈ N}, n ∈ N, of A by sets open in X in the following way: (i) Un,m = U(m) , if n = 1; (ii) Un,m = {Us(m) : s is an (n − 1)-sequence in N } ∩ Un−1,m , if n > 1. It is not hard to prove that each Un,m is an open set being the intersection of finitely many open sets, and therefore each Un is a countable increasing family of open sets in X which covers A. Since X is a set-Menger space, there is (Vn )n∈N of finite sets a sequence such that for each n, Vn ⊂ Un and A ⊂ n∈N Vn . Because each Vn is an increasing finite sequence, for each n ∈ N there is a set Un,mn ∈ Vn such that Vn = Un,mn . Therefore, we have a sequence U(m1 ) , U(m1 ,m2 ) , . . . , U(m1 ,m2 ,...,mn ) , . . . of moves by II, and the union of these sets covers A. So, σ is not a winning strategy for I. Quite similarly one proves the following two theorems. Theorem 4.22 A Lindel¨ of space X is weakly set-Menger if and only if the player I does not have a winning strategy in the game wGA for each A ⊂ X. Theorem 4.23 A Lindel¨ of space X is almost set-Menger if and only if the player I does not have a winning strategy in the game aGA for each A ⊂ X.
Acknowledgements The authors are grateful to the referee for a number of comments and suggestions which helped us to improve the exposition of the paper.
10
References [1] A.V. Arhangel’skii, An extremal disconnected bicompactum of weight c is inhomogeneous, Doklady Akademii Nauk SSSR 175 (1967), 751–754. [2] A.V. Arhangel’skii, A generic theorem in the theory of cardinal invariants of topological spaces, Commentationes Mathematicae Universitatis Carolinae 36:2 (1995), 303–325. [3] L. Babinkostova, B.A. Pansera, M. Scheepers, Weak covering properties and infinite games, Topology and its Applications 159 (2012), 3644–3657. [4] L. Babinkostova, B.A. Pansera, M. Scheepers, Weak covering properties and selection principles, Topology and its Applications 160 (2013), 2251–2271. [5] P. Daniels, Pixley-Roy spaces over subsets of the reals, Topology and its Appllications 29:1 (1988), 93–106. [6] G. Di Maio, Lj.D.R. Koˇ cinac, A note on quasi-Menger and similar spaces, Topology and its Applications 179 (2015), 148–155 [7] R. Engelking, General Topology, Revised and completed ed., Heldermann Verlag, Berlin, 1989. [8] J. Gerlits, Zs. Nagy, Some properties of C(X) I, Topology and its Applications 14:2 (1982), 151–161. ¨ [9] W. Hurewicz, Uber die Verallgemeinerung des Borelschen Theorems, Mathematische Zeitschrift 24 (1925), 401–425. [10] D. Kocev, Almost Menger and related spaces, Matematiˇcki Vesnik 61:2 (2009), 173–180. [11] Lj.D.R. Koˇ cinac, Star Menger and related spaces II, Filomat 13 (1999), 129–140. [12] Lj.D.R. Koˇ cinac, Selected results on selection principles, In: Proceedings of the Third Seminar on Geometry and Topology, July 15-17, 2004, Tabriz, Iran, 71–104. [13] P.E. Long, L.L. Herrington, Strongly θ-continuous functions, Journal of the Korean Mathematical Society 18 (1981) 21–28. ¨ [14] K. Menger, Einige Uberdeckungss¨ atze der Punktmengenlehre, Sitzungsberischte Abt. 2a, Mathematik, Astronomie, Physik, Meteorologie und Mechanik (Wiener Akademie, Wien) 133 (1924), 421–444. [15] B.A. Pansera, Weaker forms of the Menger property, Quaestiones Mathematicae 35:2 (2012), 161–169. [16] F. Rothberger, Eine Versch¨ arfung der Eigenschaft C, Fundamenta Mathematicae 30 (1938), 50–55. [17] M. Sakai, Some weak covering properties and infinite games, Central European Journal of Mathematics 12:2 (2014), 322–329. [18] M. Sakai, M. Scheepers, The combinatorics of open covers, In: K.P. Hart, J. van Mill, P. Simon (eds.), Recent Progress in General Topology III, Atlantis Press, 2014, pp. 751–800. [19] V. Saks, Ultrafilter invariants in topological spaces; abstract, Notices of the American Mathematical Society 22 (1975), A-218. [20] V. Saks, Ultrafilter invariants in topological spaces, Transactions of the American Mathematical Society 241 (1978), 79–97. [21] M. Scheepers, Combinatorics of open covers I: Ramsey theory, Topology and its Applications 69:1 (1996), 31–62. [22] Y.K. Song, R. Li, On almost Hurewicz spaces, Questions and Answers in General Topology 31 (2013), 131–136. [23] L.A. Steen, J.A. Seebach, Counterexamples in Topology, Dover Publications Inc., 1996.
11
Ljubiša D.R. Koˇcinac, Sükran ¸ Konca
PII:
S0166-8641(19)30401-8
DOI:
https://doi.org/10.1016/j.topol.2019.106996
Reference:
TOPOL 106996
To appear in:
Topology and its Applications
Received date:
24 December 2018
Revised date:
21 February 2019
Accepted date:
22 February 2019
Please cite this article as: L.D.R. Koˇcinac, S. ¸ Konca, Set-Menger and related properties, Topol. Appl. (2019), 106996, doi: https://doi.org/10.1016/j.topol.2019.106996.
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Set-Menger and related properties Ljubiˇ sa D.R. Koˇ cinac University of Niˇs, Faculty of Sciences and Mathematics, 18000 Niˇs, Serbia [email protected] S ¸u ¨ kran Konca Department of Mathematics, Bitlis Eren University, 13000 Bitlis, Turkey [email protected] Abstract In this work, we consider new types of covering properties called setMenger, set-Rothberger, set-Hurewicz and set-Gerlits-Nagy. We investigate the relationships between set-Mengerness and some other Mengertype covering properties (such as Mengerness, weak Mengerness, quasiMengerness) and study topological properties of set-Menger and related spaces. In particular, we give a game-theoretic characterization of setMenger spaces.
2010 Mathematics Subject Classification: Primary 54D20, Secondary 54B05, 54C10, 91A44 Keywords: Selection principles, game theory, Menger, set-Menger, quasi-Menger
1
Introduction
Recently, a number of papers on weaker forms of classical covering properties of Menger, Hurewicz and Rothberger has been published [3, 4, 6, 10, 11, 15, 17, 22]. In the last few years the interest to the study of weaker forms of these classical selection properties increased. Di Maio and Koˇcinac [6] have defined quasiMenger and quasi-Rothberger spaces and proved that they are different from other weak versions of Menger’s and Rothberger’s properties. In [2], Arhangel’skii defined a cardinal function sL, and spaces X such that sL(X) = ω we call sLindel¨ of : a space X is sLindel¨ of if for each subset A of X and each cover U of A by sets open in X there is a countable set V ⊂ U such that A ⊂ ∪V. Following this idea and modifying it we here consider new types of selective covering properties which will be called set covering properties. We mainly 1
investigate set-Menger and set-Rothberger properties and their relationships with other Menger-type and Rothberger-type covering properties. Throughout the paper by “a space” we mean “a topological space”. By N, R, Q and P we denote the set of natural, real, rational and irrational numbers, respectively. We use usual topological notation and terminology as in [7]. If P is a collection of subsets of a space X, we write P := {P : P ∈ P}, P := {P : P ∈ P}. If f : X → Y is a mapping between spaces X and Y , and B ⊂ Y we prefer to use f ← (B) for the preimage of B instead of f −1 (B).
2
Known definitions
Recall some definitions concerning selection principles theory and properties which will be considered in this article (see [12, 18]). A space X is said to have: 1. the Menger property M [14] (respectively, Rothberger property R [16]) if for each sequence (Un )n∈N of open covers of X there exists a sequence (Vn )n∈N (respectively, a sequence (Un )n∈N ) such that for eachn ∈ N, Vn is a finite subset of Un (respectively, Un ∈ Un , n ∈ N) and n∈N Vn = X (respectively, n∈N Un = X); 2. the Hurewicz property H if for each sequence (Un )n∈N of open covers there Vn is a finite subset of Un and each is a sequence (Vn )n∈N such that for each n, x ∈ X belongs to all but finitely many sets Vn [9]; 3. the weak Menger property wM [5] (respectively, almost Menger property asequence aM [11]) if for each sequence (Un )n∈N of open covers of X there exists (Vn )n∈N such that for each n ∈ N, Vn is a finite subset of Un and n∈N Vn = X (respectively, n∈N {V : V ∈ Vn } = X); 4. the weak Rothberger property wR [5] (respectively, almost Rothberger property aR [11]) if for each sequence (Un )n∈N of open covers of X there exists a sequence (Un )n∈N such that Un ∈ Un , n ∈ N, and n∈N Un = X (respectively, n∈N Un = X); 5. the almost Hurewicz property aH [22] (respectively, almost Gerlits-Nagy property aGN) if for each sequence (Un )n∈N of open covers of X there is a sequence (Vn )n∈N (respectively, a sequence (Un )n∈N ) such that for each n, Vn is a finite subset of Un (respectively, Un ∈ Un ) and each x ∈ X belongs to Vn (respectively, Un ) for all but finitely many n; 6. the quasi-Menger property qM [6] (respectively, quasi-Rothberger property qR [6]) if for each closed set F ⊂ X and each sequence (Un )n∈N of covers of F by sets open in X there is a sequence (Vn )n∈N (respectively, a sequence (Un )n∈N ) such that finite subset of Un (respectively, Un ∈ Un ) and for each n, Vn is a Vn ⊃ F (respectively, n∈N Un ⊃ F ); n∈N 7. the quasi-Hurewicz property qH [6] (respectively, quasi-Gerlits-Nagy property qGN [6]) if for each closed set F ⊂ X and each sequence (Un )n∈N of covers of F by sets open in X there is a sequence (Vn )n∈N (respectively, a sequence (Un )n∈N ) such that for each n ∈ N, Vnis a finite subset of Un (respectively, Un ∈ Un ) and each x ∈ F belongs to Vn (respectively, to Un ) for all but finitely many n. 2
3
New definitions
Definition 3.1 Let P be a family of nonempty subsets of a space X. We say that X is: (1) P-Menger (respectively, weakly P-Menger, almost P-Menger ) if for each A ∈ P and each sequence (Un )n∈N of sets open in X such that A ⊂ ∪Un for each n ∈ N, there is a sequence (Vn )n∈N , Vn is a finite subset of Un for each n ∈ N and A ⊂ n∈N Vn (respectively, A ⊂ n∈N Vn , A ⊂ n∈N Vn ); (2) P-Rothberger (respectively, weakly P-Rothberger, almost P-Rothberger ) if for each A ∈ P and each sequence (Un )n∈N of sets open in X such that A ⊂ Un , n ∈ N, there is a sequence (Un )n∈N such that Un ∈ Un for each n ∈ N and A ⊂ n∈N Un (respectively, A ⊂ n∈N Un , A ⊂ n∈N Un ); (3) P-Hurewicz (respectively, almost P-Hurewicz ) if for each A ∈ P and each sequence (Un )n∈N of sets open in X such that A ⊂ Un , n ∈ N, there is Un for each n ∈ N a sequence (Vn )n∈N such that Vn is a finite subset of and each x ∈ A belongs to all but finitely many sets Vn (respectively, to all but finitely many sets Vn ); (4) P-Gerlits-Nagy (respectively, almost P-Gerlits-Nagy) if for each A ∈ P and each sequence (Un )n∈N of sets open in X such that A ⊂ ∪Un , there is a sequence (Un )n∈N such that Un ∈ Un for each n ∈ N and each x ∈ A belongs to all but finitely many sets Un (respectively, all but finitely many sets Un ). To give some examples we need several definitions. The depth of a space X, denoted by g(X), is the supremum of cardinalities of closures of discrete subspaces of X [1]. Let κ be an infinite cardinal, p a filter on κ, and (xα )α∈κ a κ-sequence in a space X. A point x ∈ X is called a p-limit point of (xα )α∈κ , written plim xα = x, if for any neighbourhood U of x, {β ∈ κ : xβ ∈ U } ∈ p. Denote by μκ the set of all uniform ultrafilters on κ (p ∈ βκ \ κ is in μκ if |p| = κ). For p ∈ μκ, a space X is p-κ-compact if each κ-sequence (xα )α∈κ in X has a p-limit point in X. Let κ be an infinite cardinal number. A space X is said to be: • κ-bounded if the closure of any A ⊂ X of cardinality ≤ κ is compact (see, for example, [19, 20]); • κ-ultracompact if λ < κ and p ∈ μλ imply X is p-λ-compact [20]. Call a space X κ-σ-bounded if the closure of any A ⊂ X of cardinality ≤ κ is σ-compact.
3
Example 3.2 (1) Let P≤κ be the set of all nonempty subsets of X having cardinality ≤ κ. Then X is P≤κ -Menger provided X is κ-σ-bounded (in particular, κ-bounded). (2) Every κ-ultracompact regular space is P≤κ -Menger. (Recall that by [20, Theorem 5.4], every regular κ-ultracompact space X is κ-bounded.) (3) Let Dc be the set of all discrete closed subspaces of X. If X is g(X)bounded (or g(x)-σ-bounded), it is Dc -Menger. (4) Let L be the set of all Lindel¨of subspaces of a space X in which the closure of any Lindel¨ of subspace is σ-compact. Then X is L-Menger (in fact, L-Hurewicz). If P is the family of all nonempty subsets of X, then we say that X is setMenger (set-M) (set-Rothberger (set-R), set-Hurewicz (set-H), set-Gerlits-Nagy (set-GN)) instead of P-Menger (P-Rothberger, P-Hurewicz, P-Gerlits-Nagy). Similarly for other classes of spaces defined above. We will be concentrated on investigation of set-Menger and set-Rothberger spaces and their relatives.
4
Set-Menger and set-Rothberger spaces
We begin this section with a simple fact. Proposition 4.1 Set-Mengerness implies Mengerness, and almost set-Mengerness implies weak set-Mengerness. Remark 4.2 If X is a Menger space and P ∗ is the set of all (proper) non-dense subsets of X, then X is P ∗ -Menger. Let A be a proper non-dense subset of X and let (Un )n∈N be a sequence of open subsets of X such that A ⊂ Un for each n. For each n ∈ N, the set Vn := Un ∪{X \A} is an open coverof X.Since X is Menger, there are Wn ⊂ Vn for each n, Wn is finite and X = n∈N Wn . The sets Wn = Wn \ {X \ A}, n ∈ N, also cover A and such Wn s witness for (Un )n∈N that X is P ∗ -Menger. Similar assertions are true for set-R, set-H, set-GN properties. Theorem 4.3 Let X be a regular space. If X is almost set-Menger (almost set-Rothberger), then X is set-Menger (set-Rothberger). Proof . We will prove only the Menger case. Let A ⊂X and let (Un )n∈N be a sequence of families of open sets in X such that A ⊂ Un for each n. Since X is a regular space, for each n there exists an open cover Wn of A such that Wn := {W : W ∈ Wn } refines Un and covers A. As X is almost set-Menger, there exists (Hn )n∈N such that for each n, Hn is a finite subset of a sequence Wn and n∈N {H : H ∈ Hn } ⊃ A. For each n ∈ N, and each H ∈ Hn pick VH ∈ Un such that H ⊂ VH . Let Vn := {VH : H ∈ Hn }. Then each Vn is a finite subset of Un and A ⊂ n∈N Vn which shows that X is a set-Menger space. 4
A topological space X is a P -space if the union of countably many closed subsets of X is closed in X. Theorem 4.4 If a P -space X is weakly set-Menger, then X is almost setMenger. Proof . Let A be a subset of X and (Un )n∈N be a sequence of open covers of A by sets open in X. Since X is weakly set-Menger, there exists a sequence of (W n )n∈N such that for each n ∈ N, Wn is a finite subset Un and A ⊂ Wn . As X is a P -space, we have that the set n∈N Wn is closed. n∈N Therefore we have n∈N Wn ⊂ n∈N Wn (because theset on the left side of this inclusion is the smallest closed set containing n∈N Wn ). It follows Wn ⊂ {W : W ∈ Wn }, A⊂ n∈N
n∈N
which means that X is almost set-Menger. Corollary 4.5 Let X be a regular P -space. Then the following statements are equivalent: 1. X is set-Menger; 2. X is almost set-Menger; 3. X is weakly set-Menger. Example 4.6 There is a regular, weakly Rothberger space which is not weakly set-Rothberger. Consider the real line R equipped with the rational sequence topology τ (see [23, Example 65]): for each r ∈ Q the set {r} is open; for each irrational number x let (local) basic sets at x are of the form Uq (x) = {x} ∪ {qi : i ∈ N}, where q = (qk )k∈N is a sequence of rational numbers converging to x in the Euclidean topology. Observe that all basic sets are also closed. This space is regular [23] and weakly Rothberger as it is separable. However, this space is not weakly set-Rothberger. For, consider the subset P ⊂ R of irrational numbers . It is closed and discrete in (R, τ ). For x ∈ P and n ∈ N denote by sn (x) a sequence of rational numbers in (x − 1/2n , x) converging to x and Un (x) = sn (x) ∪ {x}. Let Un = {Un (x) : x ∈ P}. We have the sequence (Un )n∈N of covers of P = P by basic sets open in (R, τ ). If we take one Un ∈ Un , n ∈ N, then clearly P is not covered by n∈N Un . On the other hand, (R, τ ) is not Rothberger (because it is not Lindel¨of [23]), almost set-Rothberger and set-Rothberger. Theorem 4.7 Set-Mengerness implies quasi-Mengerness. Proof . Let F be a closed subset of a set-Menger space X and (Un )n∈N be a sequence of covers of F by sets open in X. Since X is set-Menger, there is finite subset of Un for each n ∈ N, and a sequence (Vn )n∈N such that Vnis a F ⊂ n∈N Vn . Then F = F ⊂ n∈N Vn , i.e. X is quasi-Menger. 5
Theorem 4.8 Quasi-Mengerness implies weak set-Mengerness. Proof . Let A be a subset of a quasi-Mengerspace X and let (Un )n∈N be a sequence of open subsets of X such that A ⊂ Un for each n. Apply the fact that X is quasi-Menger to the closed set A and the sequence (Un )n∈N . There is a sequence (Vn )n∈N such thatVn isa finite subset of Un for each n, and A ⊂ n∈N Vn . Therefore, A ⊂ n∈N Vn , i.e. X is weakly-set-Menger. In a similar way we prove the corresponding theorems for other considered classes of spaces. Therefore, we have the following diagram. set−R ⇑ set−GN ⇓ set−H ⇓ set−M ⇑ set−R
⇒ ⇒ ⇒ ⇒ ⇒
R ⇑ GN ⇓ H ⇓ M ⇑ R
⇒
qR ⇒ ⇑ ⇒ qGN ⇒ ⇓ ⇒ qH ⇒ ⇓ ⇒ qM ⇒ ⇑ ⇒ qR ⇒
w−set−R ⇓
⇐
⇓ ⇓ w−set−M ⇐ ⇑ w−set−R ⇐
a−set−R ⇑ a−set−GN ⇓ a−set−H ⇓ a−set−M ⇑ a−set−R
Diagram 1 Example 4.9 There are quasi-Rothberger (so weakly set-Rothberger) spaces which are not set-Rothberger. (1) Let RS be the Sorgenfrey line. By Proposition 2.2 in [6] it is known that RS is quasi-Rothberger, hence weakly set-Rothberger. To show that it is not set-Rothberger, let A = (0, 2) ⊂ RS . Let Un := basic sets in RS having length less than 21n and covering A = [0, 2) . If we take any Un ∈ Un , n ∈ N, then the length of all chosen elements is less than 1 1 1 + + . . . + n + . . . = 1. 2 22 2 But the length of A is 2, so A cannot be covered by selected sets Un , n ∈ N. (2) It is known that a compact space is Rothberger if and only if it is scattered. So, any hereditarily separable compact non-scattered space is weakly set-Rothberger (being quasi-Rothberger), but not set-Rothberger because it is not Rothberger. For example, the closed unit interval I = [0, 1], the Cantor set 2ω and the Hilbert cube I ω are not set-Rothberger, but they are weakly set-Rothberger. (3) According to [17, Theorem 3.4] every hereditarily Lindel¨ of space with ortho-base is hereditarily weakly Rothberger and so weakly set-Rothberger. In particular, every ccc GO space is weakly set-Rothberger; for example, the Souslin line. 6
Example 4.10 The deleted radius topology in the plane [23, Example 77] is not almost Rothberger [3], so it is not almost set-Rothberger. But it is weakly set-Menger. Example 4.11 Set-Mengerness is not a hereditary property. Let R be the real line with the usual metric topology. Assume that A isany subset of R and (Un )n∈N is a sequence of sets open in R satisfying A ⊂ Un for each n ∈ N. Since A is a Menger space as a closed subspace of the Menger space R, there are finite sets Vn ⊂ Un , n ∈ N, such that A ⊂ A ⊂ n∈N Vn . It follows that R is a set-Menger space. Consider the space P of irrational real numbers. It is known that P is not Menger, so it cannot be set-Menger. Remark 4.12 The space P is an example of a weakly set-Menger space (because according to [6, Proposition 2.2] it is quasi-Menger) which is neither setMenger nor almost set-Menger (because it is not almost Menger). Theorem 4.13 If A is a clopen (closed and open) subset of a set-Menger (setRothberger) space X, then A is also set-Menger (set-Rothberger). Proof . Let X be a set-Rothberger space and let B be a subset of (A, τA ) and (Un )n∈N be a sequence of covers of ClτA (B) (the closure of B with respect to the topology τA ) by sets open in (A, τA ). Since A is open, each Un is a family of sets open in (X, τ ). Since A is closed, ClτA (B) = Clτ (B). B is a subset of X, and (Un )n∈N is a sequence of covers of Clτ (B) by sets open inX. Because X is set-Rothberger, there exist Un ∈ Un , n ∈ N, such that B ⊂ n∈N Un . As each Un is open in (A, τA ), it follows that A is set-Rothberger. Quite similarly we prove the set-Menger part of the theorem. Theorem 4.14 Continuous image of a set-Menger (set-Rothberger) space is also set-Menger (set-Rothberger). Proof . We consider only the set-Rothberger case. Assume that X is a setRothberger space and f : X → Y is a continuous mapping from X onto Y . Let B be any subset of Y and (Vn )n∈N be a sequence of covers of B by sets open in Y . Let A = f ← (B). Since f is continuous, for each n ∈ N, f ← (Vn ) = Un is the collection of open sets in X with Vn = Un . A = f ← (B) ⊂ f ← (B) ⊂ f ← ← As X is set-Rothberger, there are Un = f (Vn ) ∈ Un , n ∈ N, such that A ⊂ n∈N Un . So f (Un ) = Vn B = f (A) ⊂ n∈N
n∈N
and we conclude that Y is set-Rothberger. In a similar way, with a small changes in the proof, we get: 7
Theorem 4.15 A continuous image of a weakly (almost) set-Menger space is also weakly (almost) set-Menger. Recall that a mapping f : X → Y is called strongly θ-continuous [13] if for each x ∈ X and each open set V containing f (x) there is an open set U containing x such that f (Cl(U )) ⊂ V . Every strongly θ-continuous mapping is continuous. Theorem 4.16 A strongly θ-continuous image Y = f (X) of an almost setMenger space X is a set-Menger space. Proof . Let B be any subset of Y and let (Vn )n∈N be a sequence of covers of B by sets open in Y . Let A = f ← (B) and let x ∈ A. Since strongly θcontinuous mappings are continuous, we have f (x) ∈ f (f ← (B)) ⊂ f (f ← (B)) = B. Thus for each n ∈ N there is a set V (x, n) ∈ Vn containing f (x). Since f is strongly θ-continuous, there is an open set U (x, n) ⊂ X containing x such that f (U (x, n)) ⊂ V (x, n). Therefore, for each n, the set Un = {U (x, n) : x ∈ A} is an open cover of A. As X is almost set-Menger, there is a sequence n )n∈N (F of finite sets such that for each n, Fn is a subset of Un and A ⊂ n∈N {F : F ∈ Fn }. To each F ∈ Fn assign a set WF ∈ Vn with f (F ) ⊂ WF and put Wn := {WF : F ∈ Fn }. We obtain the sequence (Wn )n∈N of finite subsets of Vn , n ∈ N, which witnesses for (Vn )n∈N that Y is a set-Menger space. What about pre-images of set-Menger spaces? We need a new concept called nearly set-Menger space. Recall that a subset Y of a space X is said to relatively )n∈N of open covers of X, there exist finite Menger in X if for each sequence (Un sets Vn ⊂ Un , n ∈ N, such that n∈N Vn ⊃ Y . A space X is nearly set-Menger if each subset A of X is relatively Menger in X. Theorem 4.17 The perfect preimage of a set-Menger space is nearly set-Menger. Proof . Let f : X → Y be a perfect mapping from a space X onto a setMenger space Y . Let (Un )n∈N be a sequence of open covers of X and A ⊂ X. Then B = f (A) is a subset of Y . Let y ∈ f (A). Then f ← (y) is compact in X so that for each n ∈ N there exists a finite subset Vn,y of Un such that f ← (y) ⊂ ∪Vn,y = Vn,y . Because f is closed, there is an open set Wn,y ⊂ Y such that y ∈ Wn,y and f ← (y) ⊂ f ← (Wn,y ) ⊂ Vn,y . For each n ∈ N the set Wn := {Wn,y : y ∈ f (A)} is a cover of f (A) by sets open in Y . Since Yis set-Menger, for each n there is a finite Hn ⊂ Wn such that f (A) ⊂ n∈N Hn . For each H ∈ Hn there is the corresponding finite Vn,H ⊂ Un . The set Gn := {V : V ∈ Vn,H , H ∈ Hn } is a finite subset of Un . Further, A ⊂ f ← (f (A)) ⊂ Gn . n∈N
This completes the proof.
8
Corollary 4.18 The product X × Y of a set-Menger space X and a compact Hausdorff space Y is a nearly set-Menger space. The corollary follows from the fact that the projection mapping X × Y → X is perfect, because Y is compact. Theorem 4.19 If f : X → Y is an open, perfect mapping from a space X onto a weakly set-Menger space Y , then for each set A ⊂ X and each sequence (Un )n∈N of open covers of f ← (f (A)) by sets open in X there is a sequence (Vn )n∈N of finite sets such that for each n, Vn ⊂ Un and n∈N Vn ⊃ f ← (f (A)) ⊃ A. Proof . Let A be a subset of X and (Un )n∈N a sequence of covers of f ← (f (A)) = (because f is open and continuous) by sets open in X. For each y ∈ f (A) the set Cy := f ← (y) is compact so that for each n ∈ N there is Vn (y). As f is a finite set Vn (y) ⊂ Un which covers Cy . Let Vn (y) = a closed mapping, for each n ∈ N and each y ∈ f (A) there is an open set Wn (y) ⊂ Y such that y ∈ Wn (y) and f ← (Wn (y)) ⊂ Vn (y). For each n ∈ N set Wn = {Wn (y) : y ∈ f (A)}. Then each Wn is a cover of f (A) by sets open in Y . Since Y is weakly set-Menger, there is a sequence (Hn )n∈N such that Hn is a finite subset of Wn , n ∈ N, and f (A) ⊂ n∈N Hn . For each UH . If n and each H ∈ Hn there is a finite UH ⊂ Un with f ← (H) ⊂ Gn = {U ∈ Un : U ∈ UH , H ∈ Hn }, then Gn is a finite subset of Un for each n. Since f is open, we have Hn ) = f ← ( Hn ) ⊂ Gn f ← ( Hn ) ⊂ A ⊂ f ← (f (A)) ⊂ f ← ( f ← (f (A))
n∈N
n∈N
n∈N
n∈N
which completes the proof. We end the paper by game-theoretic characterizations of set-Menger-type covering properties. Because the classical Menger property has a game-theoretic characterization it is natural to ask: Problem 4.20 Do set-Menger properties have game-theoretic characterizations? The following game is associated to the set-Menger (respectively, weakly setMenger, almost set-Menger) property. Let A be a fixed subset of a space X. In the game GA (respectively, wGA , aGA ) two players, I and II, play a round for each n ∈ N. In then − th round player I chooses a collection Un of open sets in X such that A ⊂ Un , and II responds by choosing a finite Vn subset of Un . II wins a play U1 , V1 ; U2 , V2 ; . . . , Un , Vn , . . . if A ⊂ n∈N Vn (respectively, A ⊂ n∈N Vn , A ⊂ n∈N Vn ); otherwise I wins. Notice that if I does not have a winning strategy in the game GA (respectively, wGA , aGA ) for each A ⊂ X, then X is set-Menger (respectively, weakly setMenger, almost set-Menger). The next theorems show that the converse is also true. 9
Theorem 4.21 A Lindel¨ of space X is set-Menger if and only if the player I does not have a winning strategy in the game GA for each A ⊂ X. Proof . Let A be a subset of a Lindel¨ of set-Menger space X. Let σ be a strategy that the first move of I is a set σ(∅) = U1 of sets for I in the game GA . Assume open in X such that A ⊂ U1 . One may suppose that U1 = {U(1) ⊂ U(2) ⊂ . . .}. Let for each n ∈ N, σ(U(n) ) be the set {U(n,1) ⊂ U(n,2) ⊂ U(n,3) ⊂ . . .}. We continue in this way and for each m-sequence (n1 , . . . , nm ) of natural numbers we define σ(U(n1 ) , U(n1 ,n2 ) , . . . , U(n1 ,n2 ,...,nm ) ) = {U(n1 ,...,nm ,1) ⊂ U(n1 ,...,nm ,2) ⊂ . . .}. This procedure led to the family {Us : s is a finite sequence in N } such that {U(s,1) , U(s,2) , . . .} is a cover of A by sets open in X. We define now covers Un = {Un,m : m ∈ N}, n ∈ N, of A by sets open in X in the following way: (i) Un,m = U(m) , if n = 1; (ii) Un,m = {Us(m) : s is an (n − 1)-sequence in N } ∩ Un−1,m , if n > 1. It is not hard to prove that each Un,m is an open set being the intersection of finitely many open sets, and therefore each Un is a countable increasing family of open sets in X which covers A. Since X is a set-Menger space, there is (Vn )n∈N of finite sets a sequence such that for each n, Vn ⊂ Un and A ⊂ n∈N Vn . Because each Vn is an increasing finite sequence, for each n ∈ N there is a set Un,mn ∈ Vn such that Vn = Un,mn . Therefore, we have a sequence U(m1 ) , U(m1 ,m2 ) , . . . , U(m1 ,m2 ,...,mn ) , . . . of moves by II, and the union of these sets covers A. So, σ is not a winning strategy for I. Quite similarly one proves the following two theorems. Theorem 4.22 A Lindel¨ of space X is weakly set-Menger if and only if the player I does not have a winning strategy in the game wGA for each A ⊂ X. Theorem 4.23 A Lindel¨ of space X is almost set-Menger if and only if the player I does not have a winning strategy in the game aGA for each A ⊂ X.
Acknowledgements The authors are grateful to the referee for a number of comments and suggestions which helped us to improve the exposition of the paper.
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