Pointless proofs of the Menger and Rothberger games
aa r X i v : . [ m a t h . GN ] F e b POINTLESS PROOFS OF THE MENGER AND ROTHBERGERGAMES
RENAN M. MEZABARBA
Abstract.
This work presents proofs for Hurewicz’s and Pawlikowski’s The-orems about topological games in the context of lattices, in which one cannotargue with points.
Introduction
The framework of topological games and selection principles, the latter intro-duced by Scheepers [9], have been a fruitful source for both questions and answersin the field, as can be seen in the surveys [10, 12]. We refer the reader to the arti-cle [1], by Aurichi and Dias, which presents an introduction to the subject, as wellas the main notations we follow along this work.In an attempt to extend some results regarding selective variations of tightnessin the context of C p -theory [2] to the realm of topological groups, the following twoimportant theorems about open covers appeared to be relevant: • Hurewicz’s Theorem [4].
A topological space satisfies S fin ( O , O ) if, andonly if, Player I does not have a winning strategy in the game G fin ( O , O ); • Pawlikowski’s Theorem [7].
A topological space satisfies S ( O , O ) if,and only if, Player I does not have a winning strategy in the game G ( O , O ).Roughly speaking, our original problem is to settle the same equivalences statedabove but replacing the family O , of open covers of a topological space, with theset Ω e whose elements are the subsets A of a topological group G such that e ∈ A .Although Hurewicz’s and Pawlikowski’s Theorems are not directly related to thisproblem, their proofs could provide useful insights in the search for an answer. Thisis where the work of Szewczak and Tsaban [11] enters: they presented conceptualproofs for both the results, in such a way that the role of each hypothesis becomesvery clear.Even though the analysis made so far about their proofs has not helped in ouroriginal problem, we have come to an unwittingly disclosure: both proofs can becarried out in the topology of the space rather than in the space itself, meaningthat the results are of an order-theoretic nature. In other words, Hurewicz’s andPawlikowski’s Theorems can be stated and proved in the context of Pointless Topol-ogy , in the sense of Johnstone [5, 6]. The purpose of this work is to present thisgeneralization, based on the proofs in [11] and in a few basic concepts of pointlesstopology, which we shall recall along the next sections . Mathematics Subject Classification.
Key words and phrases. topological games, Menger game, Rothberger game, lattices, pointlesstopology. For a more systematic treatment of pointless topology, we refer the reader to [5, 8]. The Menger game played on nice lattices
Let ( P , ≤ ) be a lattice , i.e., a poset such that for every a, b ∈ P there are sup { a, b } and inf { a, b } , usually denoted by a ∨ b and a ∧ b , respectively the join and the meet of a and b . For a fixed element p ∈ P , let us denote by V p the family of all subsets A of P such that sup A = p .The family V p may be thought as a generalization of the family of open coversof a topological space. Indeed, if ( X, τ ) is a topological space, then ( τ, ⊆ ) is alattice such that U ⊆ τ is an open cover for X if, and only if, sup U = X . Thus,with the notation of the previous paragraph, the family O X of the open covers of X coincides with the family V X . In this sense, the main task of this section is todiscuss the proof of the following theorem, which is a generalization of Hurewicz’sTheorem. Theorem 1.1.
Let P be a lattice with enough prime elements. For every element p ∈ P , S fin ( V p , V p ) holds if, and only if, I G fin ( V p , V p ) holds. Here, a prime element of the lattice P is an element q ∈ P (not equal to 1if P is bounded), such that a ∧ b ≤ q implies a ≤ q or b ≤ q for every a, b ∈ P .The enoughness means that if a, b ∈ P satisfy a b , then there exists a primeelement q ∈ P such that b ≤ q and a q . These elements shall play the role ofthe (complement of) points in our adaptation of the arguments of Szewczak andTsaban.Let us begin by noticing that if S fin ( V p , V p ) holds, then for every A ∈ V p thereexists a countable subset B ⊆ A such that sup B = p , i.e., P is Lindel¨of in anorder-theoretic sense. Thus we may restrict
Player II ’s moves to countable subsetsof P . On the other hand, since suprema are associative, we may assume Player I ’smoves are countable and increasing, in such a way that
Player II may select a singleelement per inning.Indeed, if
Player I plays a subset A n ∈ V p in the inning n , then there exists acountable subset { a nm : m ∈ ω } ⊆ A n such that sup m ∈ ω a nm = p , implying that B n = { b m : m ∈ ω } ∈ V p is countable and increasing, where b m = a n ∨ . . . ∨ a nm foreach m . Now, if Player II selects b m n ∈ B n for each n such that sup n ∈ ω b m n = p ,the associativity of joins yields p = sup n ∈ ω b m n = sup n ∈ ω [ (cid:8) a n , . . . , a nm n (cid:9) , showing that there is no loss of generality in our assumption. As a last remark,notice that we may suppose that an answer A = { a n : n ∈ ω } ∈ V p of Player I tosome element a ∈ P selected by Player II is such that a = a , since we can replace A with { a ∨ a n : n ∈ ω } ∈ V p .With these assumptions, a strategy for Player I in the game G fin ( V p , V p ) may beidentified with a family σ = { A s : s ∈ ω <ω } of subsets of P such that A s ∈ V p forevery finite sequence s ∈ ω <ω , where A h i = { a j : j ∈ ω } is Player I ’s first move, A h m i = { a m,j : j ∈ ω } is Player I ’s answer to a m , A h m,n i = { a m,n,j : j ∈ ω } is Player I ’s answer to a m,n and so on. Let us say that such a strategy σ is a nicestrategy for Player I in the game G fin ( V p , V p ).All the previous simplifications were made by Szewczak and Tsaban [11] in thecontext of open covers in order to reduce the proof of Hurewicz’s Theorem to a tricky Actually, it is more than a lattice: it is a frame, as we shall recall soon.
OINTLESS PROOFS OF THE MENGER AND ROTHBERGER GAMES 3 lemma about tail covers. Recall that a countable open cover U of a topological space X is called a tail cover [11] if the set of intersections of cofinite subsets of U is anopen cover for X . Here, the translation is straightforward: a subset A ⊆ P is a tail set (with respect to p ∈ P ) if sup A = p and for every cofinite subset B of A there exists inf B such that sup { inf B : B is a cofinite subset of A } = p . Theorder-theoretic version of their lemma is the following. Lemma 1.2.
Let σ = { A s : s ∈ ω <ω } be a nice strategy for Player I and for n ∈ ω let B n +1 = S s ∈ ω n A s . Then B n +1 is a tail set with respect to p . Note that B n +1 is the union of all possible answers of Player I in the n -th inning. Proof.
We proceed by induction.Since B = A h i is countable and increasing, it follows easily thatsup (cid:8) inf B : B is a cofinite subset of A h i (cid:9) = p, showing that B is a tail set. Now, we assume B n is a tail set for n >
1. Sinceevery A s is countable, we may write B n = { b n : n ∈ ω } and then we put B n +1 = [ j ∈ ω (cid:8) b jn : n ∈ ω (cid:9) , where { b m = b m , b m , b m , . . . } is the (increasing) answer of Player I to b m ∈ B n . Weshall show that B n +1 is a tail set with respect to p .Let B ⊆ B n +1 be a cofinite subset of B n +1 and for every k ∈ ω consider m k =min { n : b kn ∈ B } . Note that since B is cofinite, m k = 0 for all but finitely many k ,hence C = { k : m k = 0 } is cofinite. Now, the niceness of strategy σ implies that D = { b km k : k ∈ C } is also a cofinite subset of B n , while the induction hypothesisguarantees the existence of inf D . Finally, since inf B ∩ { b kn : n ∈ ω } = b km k forevery k ∈ ω , it follows thatinf B = inf [ k ∈ ω B ∩ (cid:8) b kn : n ∈ ω (cid:9) = inf k ∈ ω b km k = inf D ∧ ^ k C b km k , which existence follows because P is a lattice. Now, we show that the supremum ofthose cofinal subsets is p .Clearly we have inf B ≤ p for all cofinite subsets B ⊆ B n +1 . So, we just needto show that if q ∈ P is such that inf B ≤ q for all cofinite subsets B of B n +1 ,then p ≤ q . Suppose, contrary to our claim, that p q . Since P has enoughprime elements, there exists a ˜ q ∈ P such that q ≤ ˜ q and p ˜ q . Now, the inductivehypothesis yields a cofinite subset C ⊆ B n such that inf C ˜ q , say C = { b n : n ∈ J } for a cofinite subset J ⊆ ω . Since sup { b mk : k ∈ ω } = p for all m ∈ ω , for each m J there exists an n m ∈ ω such that b mn m ˜ q , from which it follows that V m ∈ ω \ J b mn m ˜ q . Since the family E := { b nm : n ∈ J, m ∈ ω } ∪ (cid:8) b km : m ∈ ω \ J and k ≥ n m (cid:9) is a cofinite subset of B n +1 such that inf E = inf C ∧ V m ∈ ω \ J b mn m , the primenessof ˜ q gives inf E ˜ q . Therefore, inf E q . (cid:3) Proof of Theorem 1.1.
The simplifications made so far allows us to assume
Player I has a nice strategy σ such that, with the same notations of the previous lemma,the family B n +1 is a tail set for every n ∈ ω . Consequently, V n +1 := { inf B : B ⊆ B n +1 is cofinite } ∈ V p . R. M. MEZABARBA
Now, S fin ( V p , V p ) applies to the sequence ( V , V , . . . ), giving a sequence ( F , F , . . . )of finite subsets F n ⊆ V n such that sup S n ∈ ω F n = p . Notice that for each f ∈ F n there exists a cofinite subset B nf ⊆ B n +1 such that f = inf B nf , and C n = T f ∈ F n B nf is a cofinite subset of B n +1 .Finally, let us see how Player II can defeat the nice strategy σ : in the n -th inning, Player I selects an infinite subset A of B n +1 , which must intersect C n , so Player II may choose f n ∈ C n ∩ A . Then p = sup { inf B nf : f ∈ F n , n ∈ ω \ { }} ≤ sup n ∈ ω f n ≤ p, showing that Player II wins, as desired. (cid:3) The Rothberger game played on nicer lattices
We now proceed to discuss the adaptation of Pawlikowski’s Theorem, which shallrequire stronger conditions on the lattice P . For our preliminary considerations, letus suppose that P is a bounded lattice with enough prime elements. For simplicity,such lattices will be called pre-Pawlikowski .As it happens in the original proof of Pawlikowski, we will need to use Hurewicz’stheorem to provide nice plays in auxiliary games. At the beginning of the proof,one uses the simple exercise that a countable union of Menger spaces is again aMenger space. This still happens with pre-Pawlikowski lattices when we considerthe right generalization of union. Lemma 2.1.
If for every n ∈ ω there is a pre-Pawlikowski lattice P n such that S fin ( V P n , V P n ) holds, then P := Q n ∈ ω P n is a pre-Pawlikowski lattice satisfying S fin ( V P , V P ) , where P is endowed with the pointwise ordering and P := (1 P n ) n ∈ ω .Proof. The reader can easily show that P is a pre-Pawlikowski lattice such that 1 P is its maximum element. Now, for a sequence ( A n ) n ∈ ω of subsets of P such thatsup A n = 1 P for each n ∈ ω , we shall select a sequence ( F n ) n ∈ ω of subsets such that F n ∈ [ A n ] <ω for every n and sup S n ∈ ω F n = 1 P .Let { Q n : n ∈ ω } be an infinite partition of ω such that each Q n is infinite. Fora fixed n ∈ ω and j ∈ Q n we have sup A j = 1 P , implying that sup π n [ A j ] = 1 P n ,where π n : P → P n is the obvious projection. Then, for each j ∈ Q n there exists afinite subset F j ⊆ A j such that sup S j ∈ ω π n [ A j ] = 1 P n , from which it easily followsthat sup S n ∈ ω F n = 1 P . (cid:3) With this lemma, we can show that if P is a pre-Pawlikowski lattice satisfying S fin ( V , V ), then every strategy of Player I in the game G fin ( V , V ) can be severelydefeated, in the following sense. Proposition 2.2.
Let P be a pre-Pawlikowski lattice and let σ be a strategy for Player I in the game G fin ( V , V ) . If S fin ( V , V ) holds, then there exists a play ( A , F , A , F , . . . ) according to σ such that for every p ∈ P \ { } we have sup F n p for infinitely many n .Proof. The previous lemma guarantees that P ω is a pre-Pawlikowski lattice satis-fying S fin ( V ω , V ω ), where 1 ω := (1) n ∈ ω . Now, we use σ to cook up a strategy˜ σ for Player I in the game G fin ( V ω , V ω ) played on P ω . For a subset A ⊆ P , let˜ A := { δ n ( a ) : a ∈ A and n ∈ ω } , where δ n ( a ) ∈ P ω is such that δ n ( a )( m ) := 0 for m = n and δ n ( a )( n ) = a . It is easy to see that if sup A = 1, then sup ˜ A = 1 ω . OINTLESS PROOFS OF THE MENGER AND ROTHBERGER GAMES 5
We define ˜ σ ( ∅ ) := ˜ A , where σ ( ∅ ) := A . If Player II selects a finite subset˜ F ⊆ ˜ A , let F := { a ∈ A : δ n ( a ) ∈ ˜ F for some n ∈ ω } , a finite subset of A ,and then define ˜ σ ( ˜ F ) := ˜ A , where A := σ ( F ). Proceeding like this, we obtain astrategy ˜ σ for Player I in the game G fin ( V ω , V ω ) played on P ω .The lattice version of Hurewicz’s theorem yields a play ( ˜ A , ˜ F , ˜ A , ˜ F , . . . ) ac-cording to ˜ σ such that sup S n ∈ ω ˜ F n = 1 ω , and by the way we defined ˜ σ , thisplay corresponds to a play ( A , F , A , F , . . . ) according to σ . We claim thislatter play has the desired property. Indeed, for a fixed p ∈ P \ { } , we have˜ p := ( p, , , . . . ) < ω , from which it follows that there exists n ∈ ω and f :=( f n ) n ∈ ω ∈ ˜ F n such that f ˜ p , implying f ∈ F n is such that f p , hencesup F n p . Since F := S j ≤ n F j is finite, the set N := { n : δ n ( a ) ∈ F for some a } is also finite, which allow us to take ˜ p m := (1 , . . . , , p, , . . . ), with p in the m -thposition, for m N . Once again, there exists n ≥ m > n such that sup F n p .Continuing like this, we see that sup F n p for infinitely many n . (cid:3) In the argument presented by Szewczak and Tsaban [11], the topological coun-terpart of Proposition 2.2 is used to obtain a play ( U , F , U , F , . . . ) of an auxiliaryinstance of the Menger game such that for every x one has x ∈ S F n for infinitelymany n . This allows one to select U m ∈ F m for each m ∈ ω such that the family { U m : m ∈ ω } is an open cover for X . As it happens with the previous propositions,this argument also has a lattice counterpart. Lemma 2.3.
Let P be a pre-Pawlikowski lattice satisfying S ( V , V ) . Let ( F n ) n ∈ ω be a sequence of finite subsets of P such that for every p ∈ P \ { } there are infinitelymany n such that sup F n p . Then there are elements f n ∈ F n for every n suchthat sup n ∈ ω f n = 1 .Proof. The proof is the same as in [11], up to terminology. For each n ∈ ω let V n := n^ G : ∃ J ∈ [ ω ] n +1 ∀ j ∈ J | G ∩ F j | = 1 and | G | ≤ n + 1 o , i.e., the elements of V n are the meets of n + 1 elements belonging to n + 1 pairwisedistinct sets of the sequence ( F j ) j ∈ ω . The hypothesis about the sequence ( F n ) n ∈ ω implies that sup V n = 1 for every n ∈ ω : for if p <
1, there exists a prime element˜ p < p ≤ ˜ p , while the hypothesis gives infinitely many m such thatsup F m ˜ p ; then we may take J ∈ [ ω ] n +1 and f j ∈ F j for each j ∈ J such that f j ˜ p ; since ˜ p is prime, it follows that V j ∈ J f j ˜ p , hence V j ∈ J f j p . Therefore,1 ∈ P is the only upper bound of V n .Now, since S ( V , V ) holds, there exists a sequence ( v n ) n ∈ ω with v n ∈ V n foreach n , such that sup n ∈ ω v n = 1. Notice that v ≤ f n for some f n ∈ F n , while v ≤ f n for some f n ∈ F n such that n = n , and so on. Proceeding like this, weobtain f n k ∈ F n k such that sup k ∈ ω f n k = 1, so we can pick arbitrary f n ∈ F n forthe remaining n ’s and still get sup n ∈ ω f n = 1. (cid:3) We finally proceed to the order-theoretic version of Pawlikowski’s theorem. How-ever, we must impose a last condition over our lattices. Let us say that P is a Pawlikowski lattice if (sup A ) ∧ b = sup { a ∧ b : a ∈ A } holds for every b ∈ P andevery subset A ⊆ P having a supreme. This is the order-theoretic version of theidentity (cid:16)[ U (cid:17) ∩ B = [ U ∈U U ∩ B, R. M. MEZABARBA holding for (open) sets of a topological space X . The distributivity over arbitrarysuprema is needed to guarantee that for every A, B ∈ V , the family A ∧ B := { a ∧ b : a ∈ A and b ∈ B } is still a member of V , i.e., sup A ∧ B = 1. Theorem 2.4.
Let P be a Pawlikowski lattice satisfying S ( V , V ) . Then Player I does not have a winning strategy in the game G ( V , V ) .Proof. The hypothesis over P allows one to follow the same arguments of Szewczakand Tsaban [11] by simply applying the correspondent order-theoretic versions ofthe necessary lemmas. The details are left to the reader. (cid:3) Further comments
Although the adaptations of Szewczak and Tsaban simplifications do not requireexplicit use of points, one could ask how the further assumptions on Pawlikowskilattices affect their proximity to topological spaces. Recall that a lattice P is spatial if there exists a topological space ( X, τ ) such that P is a frame isomorphic to τ inthe frame category , where by a frame we mean a complete lattice satisfying thedistributive condition of Pawlikowski lattices. Thus, the question rephrases as: isthere a non-spatial Pawlikowski lattice? The answer is yes.Indeed, as implicitly showed by Johnstone in his classic book Stone Spaces [5], alattice P is spatial if, and only if, P is a complete lattice with enough prime elements.On the other hand, our assumption regarding distributivity and prime elements donot yield completeness, as the following example of Eric Worsey shows . Example 3.1.
For an infinite set X , let L := { A ⊆ X : A is finite or A is cofinite } .One can easily see that L is a non-complete lattice with respect to inclusion, withenough prime elements (the complementary of singletons) and such that (sup A ) ∩ B = sup A ∈A ( A ∩ B ) for every B ∈ L and A ⊆ L with sup A = S A ∈ L . In this sense, the results obtained along this work show that, although a fewaspects of points (disguised as prime elements) seem to be necessary to proveHurewicz’s and Pawlikowski’s theorems, one does not need the full power of a topol-ogy to do so. Therefore, assuming other problems regarding topological games canbe rephrased in the lattice language, it is natural to ask the following.
Question 3.2.
Do Theorems 1.1 and 2.4 hold for larger classes of lattices?
While the last question implicitly aims for possible applications, another direc-tion could be to search for generalizations of known results in the field. In this case,we mention the following theorem, which extends Lemma 3.12 from Garc´ıa-Ferreiraand Tamariz-Mascar´ua [3].
Theorem 3.3.
For a Pawlikowski lattice P and a function f : ω → ω , the followingare equivalent:(1) S ( V , V ) holds; The arrows are the increasing maps preserving arbitrary joins and finite meets, thus including0 and 1. It was pointed out by Eric Wofsey in this answer: https://math.stackexchange.com/a/2606878/128988 . See: https://math.stackexchange.com/a/4019499/128988 . OINTLESS PROOFS OF THE MENGER AND ROTHBERGER GAMES 7 (2) for each sequence ( A n ) n ∈ ω where A n ∈ V for every n , there are finitesubsets B n ∈ [ A n ] I wish to express my gratitude to Dione Andrade Lara and Rodrigo “Rockdays”Dias for the fruitful discussions about the results presented here. References [1] L. F. Aurichi and R. R. Dias. A minicourse on topological games. Topology and its Applica-tions , 258:305–335, 2019.[2] L. F. Aurichi and R. M. Mezabarba. Bornologies and filters applied to selection principlesand function spaces. Topology and its Applications , 258:187–201, 2019.[3] S. Garc´ıa-Ferreira and A. Tamariz-Mascar´ua. Some generalizations of rapid ultrafilters intopology and id-fan tightness. Tsukuba Journal of Mathematics , 19(1):173–185, 1995.[4] W. Hurewicz. ¨Uber eine Verallgemeinerung des Borelschen Theorems. MathematischeZeitschrift , 24(1):401–421, 1926.[5] P. T. Johnstone. Stone Spaces . Cambridge University Press, 1 edition, 1982.[6] P. T. Johnstone. The point of pointless topology. Bull. Amer. Math. Soc. (N.S.) , 8(1):41–53,01 1983.[7] J. Pawlikowski. Undetermined sets of point-open games. Fundamenta Mathematicae ,144(3):279–285, 1994.[8] J. Picado and A. Pultr. Frames and Locales: Topology Without Points . Frontiers in Mathe-matics. Springer Basel, 2012.[9] M. Scheepers. Combinatorics of open covers I: Ramsey theory. Topology and its Applications ,69(1):31–62, 1996.[10] M. Scheepers and M. Sakai. The combinatorics of open covers. In K. P. Hart, J. van Mill,and P. Simon, editors, Recent Progress in General Topology III , chapter 18, pages 751–800.Atlantis, 2014.[11] P. Szewczak and B. Tsaban. Conceptual proofs of the Menger and Rothberger games. Topologyand its Applications , 272, 2020.[12] B. Tsaban. Menger’s and Hurewicz’s problems: solutions from “The Book” and refinements. Contemporary Mathematics , 533:211–226, 2011. Centro de Ciˆencias Exatas, Universidade Federal do Esp´ırito Santo, Vit´oria, ES,29075-910, Brazil Email address ::