Menger remainders of topological groups
aa r X i v : . [ m a t h . GN ] J un MENGER REMAINDERS OF TOPOLOGICAL GROUPS
ANGELO BELLA, SEC¸ ˙IL TOKG ¨OZ, AND LYUBOMYR ZDOMSKYY
Abstract.
In this paper we discuss what kind of constrains combina-torial covering properties of Menger, Scheepers, and Hurewicz imposeon remainders of topological groups. For instance, we show that such aremainder is Hurewicz if and only it is σ -compact. Also, the existenceof a Scheepers non- σ -compact remainder of a topological group followsfrom CH and yields a P -point, and hence is independent of ZFC. Wealso make an attempt to prove a dichotomy for the Menger property ofremainders of topological groups in the style of Arhangel’skii. Introduction
All topological spaces are assumed to be completely regular. All un-defined topological notions can be found in [13]. For a space X and itscompactification bX the complement bX \ X is called a remainder of X .The interplay between the properties of spaces and their remainders hasbeen studied since more than 50 years and resulted in a number of dual-ity results describing properties of X in terms of those of their remainders.A typical example of such a duality is the celebrated result of Henriksenand Isbell stating that a topological space X is Lindel¨of if and only if all(equivalently any) of its remainders is of countable type , that is, any com-pact subspace can be enlarged to another compact subspace with countableouter base.In the last years, remainders in compactifications of topological groupshave been a popular topic. This is basically due to the fact that topologicalgroups are much more sensitive to the properties of their remainders thantopological spaces in general. A major role in this study was played byArhangel ′ skii, who initiated a systematic study of this topic. Among manyother things, he obtained two elegant results, which are dichotomies fornon-locally compact topological groups. Theorem 1.1 ([5]) . Let G be a topological group. If bG is a compactificationof G , then bG \ G is either Lindel¨of or pseudocompact. Mathematics Subject Classification.
Primary: 03E75, 54D40, 54D20. Secondary:03E35, 54D30, 54D80.
Key words and phrases.
Remainder, topological group, Menger space, Hurewicz space,Scheepers space, ultrafilter, forcing.The first author’s research that led to the present paper was partially supportedby a grant of the group GNSAGA of INdAM. The second author would like to thankHacettepe University BAP project 014 G 602 002 for its support. The third authorwould like to thank the Austrian Academy of Sciences (APART Program) as well as theAustrian Science Fund FWF (Grant I 1209-N25) for generous support for this research.
Theorem 1.2 ([4]) . Let G be a topological group. If bG is a compactificationof G , then bG \ G is either σ -compact or Baire. We recall that a topological space X is Baire if the intersection of count-ably many open dense subsets is dense.In this paper we will focus our attention on topological properties whichare strictly in between σ -compact and Lindel¨of. Recall from [22] that a space X is Menger (or has the
Menger property ) if for any sequence ( U n ) n ∈ ω ofopen covers of X one may pick finite sets V n ⊂ U n in such a way that { S V n : n ∈ ω } is a cover of X . A family { W n : n ∈ ω } of subsets of X is called an ω -cover (resp. γ -cover) of X , if for every F ∈ [ X ] <ω the set { n ∈ ω : F ⊂ W n } is infinite (resp. co-finite). The properties of Scheepers and
Hurewicz are defined in the same way as the Menger property, theonly difference being that we additionally demand that { S V n : n ∈ ω } is a ω -cover (resp. γ -cover) of X . It is immediate that σ -compact ⇒ Hurewicz ⇒ Scheepers ⇒ Menger ⇒ Lindel¨of . The properties mentioned above have recently received great attention,mainly because of their combinatorial nature and game-theoretic charac-terizations. One of the most striking results about the Menger property isdue to Aurichi who proved [6] that any Menger space is a D -space.Our initial idea was to find counterparts of the properties of Menger,Scheepers, and Hurewicz in the style of Theorems 1.1 and 1.2. It turnedout that the counterpart of the Hurewicz property is already given by The-orem 1.1 because of the following result, see Section 2 for its proof. Theorem 1.3.
Let G be a topological group. If βG \ G is Hurewicz, thenit is σ -compact. Let us note that there are ZFC examples of Hurewicz sets of reals whichare not σ -compact (see [17, Theorem 5.1] or [25, Theorem 2.12]), and thusTheorem 1.3 is specific for remainders of topological groups.As it follows from the theorems below, which are the main results of thispaper, for the properties of Scheepers and Menger the situation dependson the ambient set-theoretic universe. Each subspace of P ( ω ) (e.g., anultrafilter) is considered with the subspace topology. Let us recall from [17,Theorem 3.9] that if all finite powers of a topological space X are Mengerthen X is Scheepers. The converse of this statement fails consistently: underCH there exists a Hurewicz subspace of P ( ω ) whose square is not Menger,see [17, Theorem 3.7]. Theorem 1.4.
There exists a Scheepers ultrafilter iff there exists a topo-logical group G such that βG \ G is Scheepers and not σ -compact iff thereexists a topological group G such that all finite powers of βG \ G are Mengerand not σ -compact. Corollary 1.5.
The existence of a topological group G such that βG \ G is Scheepers (resp. has all finite powers Menger) and not σ -compact isindependent from ZFC. More precisely, such a group exists under d = c ,and its existence yields P -points. ENGER REMAINDERS OF TOPOLOGICAL GROUPS 3
Theorem 1.4 and Corollary 1.5 are proved in Section 3. Let us note thatthere exists a ZFC example of a dense Baire subspace X of [ ω ] ω all of whosefinite powers are Menger (and thus also Scheepers), and hence it is indeedessential in Theorem 1.4 and Corollary 1.5 that we consider remainders oftopological groups. In fact, such a subspace X can be chosen to be a filter,see [11, Claim 5.5] and the proof of [20, Theorem 1].It is worth mentioning here that Scheepers ultrafilters have been studiedintensively under different names for decades: In [10] Canjar proved that d = c implies the existence of an ultrafilter whose Mathias forcing does notadd dominating reals. By [11, Theorem 1.1] the latter property for filterson ω is equivalent to the Menger one, and by [11, Claim 5.5] all finite pow-ers of a Menger filter are Menger. Combining this with [17, Theorem 3.9]we conclude that for filters on ω , the Menger property is equivalent to theScheepers one, and hence Scheepers (equivalently Menger [in all finite pow-ers]) ultrafilters are exactly those studied in [10]. Another descriptions ofsuch ultrafilters may be found in [9] and [15], where they were characterizedas ultrafilters with certain topological and combinatorial properties strongerthan being a P -point, respectively. In particular, there are no Scheepers ul-trafilters in models of ZFC without P -points.Regarding the Menger property, we have the following partial result es-tablished in Section 4. Note that the assumption on the remainder wemake in it is formally weaker than that made in Theorem 1.4, see the lastequivalent statement there. Theorem 1.6.
It is consistent that for any topological group G and com-pactification bG , if ( bG \ G ) is Menger, then it is σ -compact. Let us note that the properties of Menger, Scheepers, Hurewicz, havingMenger square, etc., are preserved by perfect maps in both directions. Thisimplies that if one of the remainders of a space X has one of these coveringproperties, then all others also have it, see the beginning of Section 3 formore details and corresponding definitions. Thus Theorems 1.3, 1.4, and 1.6admit several equivalent reformulations, given by the freedom to considereither all or some (specific) compactifications.In light of Theorems 1.6 and 1.4 it is natural to ask the following ques-tions. Question 1.7.
Is there a ZFC example of a topological group with a Mengernon- σ -compact remainder? Question 1.8.
Is it consistent that there exists a topological group G suchthat βG \ G is Menger and not Scheepers? Does CH imply the existence ofsuch a group?Since we do not have an analogous statement to Theorem 1.4 for theMenger property (in Theorem 1.6 we make a somewhat unpleasant assump-tion that the square of the remainder is Menger), it may still be the casethat for the Menger property there exists a dichotomy similar to Theo-rems 1.1 and 1.2. In Section 5 we analyze some properties which might becounterparts of the Menger one for remainders of topological groups. ANGELO BELLA, SEC¸ ˙IL TOKG ¨OZ, AND LYUBOMYR ZDOMSKYY Hurewicz remainders
According to the definition on [3, p. 235], a topological group G is feath-ered if it contains a non-empty compact subspace with countable outer base.Recall that a family U of open subsets of a topological space X is an outerbase for a subset A of X if A ⊂ U for all U ∈ U , and for every open O ⊃ A there exists U ∈ U such that U ⊂ O . By [3, Lemma 4.3.10] every featheredgroup has a compact subgroup with countable outer base. A topologicalgroup G is Raikov-complete if it is complete in the uniformity generated bysets { ( x, y ) ∈ G : xy − , x − y ∈ U } , where U is a neighbourhood of theneutral element of G , see [3, § Lemma 2.1.
For a feathered group G the following conditions are equiva-lent: (1) G is ˇCech-complete; (2) Each closed subgroup G of G admitting a dense σ -compact subspaceis ˇCech-complete; (3) There exists a compact subgroup H of G with countable outer basesuch that h QH i is ˇCech-complete for every countable Q ⊂ G , wherefor X ⊂ G we denote by h X i the smallest subgroup of G containing X .Proof. The implication (1) → (2) is straightforward, and (2) → (3) is adirect consequence of [3, Prop. 4.3.11]. The proof of (3) → (1) will beobtained by a tiny modification of that of [3, Theorem 4.3.15]. In particular,all details missing below can be found in the proof of the above-mentionedtheorem.Let h V n : n ∈ ω i be a decreasing sequence of open symmetric neigh-bourhoods of H such that V n +1 ⊂ V n for all n and H = T n ∈ ω V n . By [3,Lemma 3.3.10] there exists a continuous prenorm on G which satisfies { x ∈ G : N ( x ) < / n } ⊂ V n ⊂ { x ∈ G : N ( x ) < / n } for all n ∈ ω . Thus H = { x ∈ G : N ( x ) = 0 } . Let ρ be a pseudometric on G defined by ρ ( x, y ) = N ( xy − ) + N ( x − y ) for all x, y ∈ G . Then ρ is contin-uous and ρ ( x, y ) = 0 iff xy − , x − y ∈ H . Consider the equivalence relation ∼ on G defined by x ∼ y iff ρ ( x, y ) = 0 and denote by X the quotient space G/ ∼ . Let π : G → X be the quotient map and set ρ ∗ ( π ( x ) , π ( y )) = ρ ( x, y )for any x, y ∈ G . It follows that π is perfect , ρ ∗ is well-defined, and is ametric generating the quotient topology on X .The proof of [3, Theorem 4.3.15] is done as follows: Assuming that G is Raikov-complete, it is shown that ρ ∗ is complete. The same argument,applied to any subgroup G ′ of G which contains H (and hence is closedunder ∼ ) yields that if G ′ is Raikov-complete, then ρ ∗ ↾ π [ G ′ ] is complete.Our item (3) implies that h QH i is Raikov-complete for every countable Q ⊂ G because ˇCech-complete groups are Raikov-complete, see, e.g., [3, Following [3, § N : G → R + a prenorm , if N ( e ) = 0, N ( x − ) = N ( x ), and N ( xy ) ≤ N ( x ) + N ( y ) for all x, y ∈ G . ENGER REMAINDERS OF TOPOLOGICAL GROUPS 5
Theorem 4.3.7]. Therefore by (3) we have that ρ ∗ ↾ π [ h QH i ] is completefor any countable Q ⊂ G . Since π is perfect, the latter gives that ρ ∗ ↾ Y is complete for any separable closed Y ⊂ X , and hence ρ ∗ is complete.therefore G is ˇCech-complete being a perfect preimage of a complete metricspace, see [13, Theorems 3.9.10 and 4.3.26]. (cid:3) We are in a position now to present the
Proof of Theorem 1.3.
Let G be a topological group such that βG \ G is Hurewicz. Then βG \ G is Lindel¨of, hence G is of countable type [14],and therefore it is feathered. By Lemma 2.1 it is enough to show that eachclosed subgroup G of G admitting a dense σ -compact subspace is ˇCech-complete. Let G be as above, F be a dense σ -compact subspace of G , and X = G \ G , where the closure is taken in βG . Since X is closed in βG \ G ,it is Hurewicz. Applying [8, Theorem 27] to the Hurewicz space X andˇCech-complete space G \ F containing it, we conclude that there exists a σ -compact space F ′ such that X ⊂ F ′ ⊂ G \ F , which implies that G \ F ′ is a dense (because it contains F ) ˇCech-complete subspace of G . Thus G is ˇCech-complete by [5, Theorem 1.2], which completes our proof. ✷ It is well-known [27, Lemma 22] that if player II has a winning strategyin the Menger game on a space X (see Section 4 for its definition) then X is Hurewicz. Therefore Theorem 1.3 generalizes [7, Corollary 3.5].3. Scheepers remainders
In the proof of Theorem 1.4, which is the main goal of this section, weshall need set-valued maps, see [19] for more information on them. By a set-valued map
Φ from a set X into a set Y we understand a map from X into P ( Y ) and write Φ : X ⇒ Y (here P ( Y ) denotes the set of all subsetsof Y ). For a subset A of X we set Φ( A ) = S x ∈ A Φ( x ) ⊂ Y . A set-valuedmap Φ from a topological space X to a topological space Y is said to be • compact-valued , if Φ( x ) is compact for every x ∈ X ; • upper semicontinuous , if for every open subset V of Y the set Φ − ( V ) = { x ∈ X : Φ( x ) ⊂ V } is open in X .To abuse terminology, we shall call compact-valued upper semicontinuousmaps cvusc maps. It is known [28, Lemma 1] that all combinatorial coveringproperties considered in this paper are preserved by cvusc maps. Also, if f : X → Y is a perfect map, then f − : Y ⇒ X assigning to y ∈ Y thesubset f − ( y ) of X , is a cvusc maps. Therefore the properties of Menger,Scheepers, Hurewicz, having Menger square, etc., are preserved by perfectmaps in both directions. That is, if f is perfect and Z ⊂ X (resp. Z ⊂ Y ) has one of these properties, then so does f ( Z ) (resp. f − ( Z )). Inparticular, this implies that if one of the remainders of a space X has oneof these covering properties, then all others also have it. In addition, all We refer here to the online version of the paper available from the web-pages of theauthors, which is an extended version of the published one.
ANGELO BELLA, SEC¸ ˙IL TOKG ¨OZ, AND LYUBOMYR ZDOMSKYY these properties are preserved by product with ω equipped with the discretetopology .We shall also need some additional notation. For X ⊂ P ( ω ) we shalldenote by ∼ X the set { ω \ x : x ∈ X } . Note that ∼ X is homeomorphic to X because x ω \ x is a homeomorphism from P ( ω ) to itself. For subsets a, b of ω (resp. a, b ∈ ω ω ) a ⊂ ∗ b (resp. a ≤ ∗ b ) means | a \ b | < ω (resp. |{ n : a ( n ) > b ( n ) }| < ω ). A collection F of infinite subsets of ω is called a semifilter if for any a ∈ F and a ⊂ ∗ b we have b ∈ F . For a semifilter F we set F + = { x ⊂ ω : ∀ a ∈ F ( a ∩ x = ∅ ) } . Note that F + = P ( ω ) \ ∼ F . F r denotes the minimal with respect to inclusion semifilter which consistsof all co-finite sets. For the other notions used in the proof of the followingstatement we refer the reader to [3]. Lemma 3.1.
Suppose that G is a topological group, K is a compact sub-group of G with countable outer base in G , and QK is dense in G for somecountable Q ⊂ G . Let P be a property of topological spaces preserved byimages under cvusc maps and product with ω equipped with the discretetopology.If βG \ G has P and is not σ -compact, then there exists a semifilter F such that F + ⊂ F and F has P . If, moreover, ( βG \ G ) is Menger, thenthere exists a semifilter F such that F = F + and F is Menger.Proof. Observe that G is not locally compact because otherwise its remain-ders would be compact. Since G is feathered, there exists a ˇCech-completegroup ˜ G containing G as a dense subgroup, see [3, Theorem 4.3.16]. Let β ˜ G be the Stone- ˇCech compactification of ˜ G . It follows that β ˜ G \ G is not σ -compact, and hence G = ˜ G . Fix g ∈ ˜ G \ G and note that gG is dense in β ˜ G and gG ∩ G = ∅ . Therefore both β ˜ G \ G and β ˜ G \ gG have property P beingremainders of spaces homeomorphic to G , and β ˜ G = ( β ˜ G \ G ) S ( β ˜ G \ gG ).Note that K has a countable outer base also in ˜ G , and hence the quotientspace X := ˜ G/K = { zK : z ∈ ˜ G } is metrizable. It is also separable by ourassumption on G , so there exists a metrizable compactification bX of X .In addition, the quotient map π K : ˜ G → X , π K ( z ) = zK , is perfect by [3,Theorem 1.5.7], and hence by [13, Theorem 3.7.16] it can be extended to a(perfect) map π : β ˜ G → bX such that π ( β ˜ G \ ˜ G ) = bX \ X . π ↾ ˜ G = π K ,hence G = π − ( π ( G )), gG = π − ( π ( gG )), and consequently A := π ( β ˜ G \ gG )and B := π ( β ˜ G \ G ) are both co-dense subsets of bX with property P covering bX .Note that bX has no isolated points because both G = π − ( π ( G )) and β ˜ G \ G = π − ( π ( β ˜ G \ G )) are nowhere locally compact. Since bX is ametrizable compact, there exists a continuous surjective map f : P ( ω ) → X . Applying [13, 3.1.C(a)] we can find a closed subspace T of P ( ω ) suchthat f ↾ T → bX is surjective and irreducible, i.e., f [ T ′ ] = bX for anyclosed T ′ ( T . T has no isolated points: if t ∈ T were isolated then theirreducibility of f ↾ T would give that t = ( f ↾ T ) − [ f ( t )], which infers For the Scheepers property this fact is slightly non-trivial and follows from [26, Propo-sition 4.7].
ENGER REMAINDERS OF TOPOLOGICAL GROUPS 7 that f ( t ) is isolated in bX and this leads to a contradiction. Therefore T is homeomorphic to P ( ω ), and hence there exists a continuous surjectiveirreducible h : P ( ω ) → bX . Since h is irreducible, both C := h − ( A ) and D := h − ( B ) are co-dense. Since h is perfect, they also have P : both ofthem are cvusc images of β ˜ G \ G . Note also that P ( ω ) = C S D .The Cantor set P ( ω ) has the following fundamental property (see [2]and references therein) which can be directly proved by Cantor’s celebratedback-and-forth argument: for any countable dense subsets I , I , J , J of P ( ω ) such that I ∩ I = ∅ and J ∩ J = ∅ there exists a homeomorphism i : P ( ω ) → P ( ω ) with the property i ( I ) = J and i ( I ) = J . Thereforethere is no loss of generality to assume that [ ω ] <ω ⊂ P ( ω ) \ C and F r ⊂P ( ω ) \ D . Set F = { x ⊂ ω : ∃ c ∈ C, u ∈ [ ω ] <ω , v ⊂ ω ( x = ( c \ u ) ∪ v ) } , I = { x ⊂ ω : ∃ d ∈ D, u ∈ [ ω ] <ω , v ⊂ ω ( x = ( d ∪ u ) \ v ) } , and note that both F and ∼ I are semifilters. It follows that both F and I are countable unions of continuous images of C × P ( ω ) and D × P ( ω ),respectively, and consequently they are cvusc images of ( β ˜ G \ G ) × ω , where ω is considered with the discrete topology. Since the property P is preservedby product with ω , we conclude that both F and I have it.Set F = F ∪ ∼ I and note that it has property P for the same reasonas F , I do. Since C ⊂ F ⊂ F and D ⊂ I = ∼ ( ∼ I ) ⊂∼ F , we havethat P ( ω ) = F S ∼ F . Therefore F + ⊂ F because F + = P ( ω ) \ ∼ F .To prove the “moreover” part assume that F is Menger and considerthe following map φ : ( F ∩ ∼ F ) → [ ω ] ω : φ ( a ) = ( a ∪ { n + 1 : n ∈ a } ) \ a. Note that
F ∩ ∼ F is homeomorphic to the closed subset { ( x, x ) : x ∈P ( ω ) } ∩ ( F × ∼ F ) of the Menger space
F × ∼ F and thus is Menger itself.Therefore φ ( F ∩ ∼ F ) is not dominating.Every strictly increasing sequence ¯ k = ( k n ) n ∈ ω of integers such that k = 0 generates a monotone surjection ψ ¯ k : ω → ω by letting ψ − k ( n ) =[ k n , k n +1 ). We claim that there exists ¯ k as above such that ψ ¯ k ( F ) + = ψ ¯ k ( F ).Suppose to the contrary that for every ¯ k there exists a ¯ k ∈ F such that ψ ¯ k ( a ¯ k ) ∈ ψ ¯ k ( F ) \ ψ ¯ k ( F ) + , i.e., ω \ ψ ¯ k ( a ¯ k ) ∈ ψ ¯ k ( F ). Then both b ¯ k := ψ − k ( ψ ¯ k ( a ¯ k )) and ω \ b ¯ k are in F , and therefore b ¯ k ∈ F ∩ ∼ F . Note, however,that φ ( b ¯ k ) ⊂ { k n : n ∈ ω } , which means that ¯ k ≤ ∗ φ ( b ¯ k ). Since ¯ k was chosenarbitrarily we get that φ ( F ∩ ∼ F ) is dominating, which is impossible. Thiscontradiction implies that ψ ( F ) + = ψ ( F ) for some monotone surjection ψ : ω → ω , and then ψ ( F ) is the semifilter with Menger square we werelooking for. (cid:3) Recall that a family
X ⊂ P ( ω ) is centered if T X ′ is infinite for every X ′ ∈ [ X ] <ω . We are in a position now to present the Proof of Theorem 1.4. If F is a Scheepers ultrafilter, then ∼ F is a subgroupof ( P ( ω ) , ∆) and F ∪ ∼ F = P ( ω ). Thus F is a Scheepers non- σ -compact ANGELO BELLA, SEC¸ ˙IL TOKG ¨OZ, AND LYUBOMYR ZDOMSKYY (no ultrafilter can be Borel) remainder of the group ( ∼ F , ∆). Moreover, allfinite powers of F are Menger (and hence also Scheepers) by [11, Claim 5.5].Let us now prove the “if” part, i.e., assume that ( βG \ G ) is Scheepersand not σ -compact. In the same way as at the beginning of the proofof Theorem 1.3 we conclude that G is feathered. By Lemma 2.1 we mayassume without loss of generality that G satisfies the premises of Lemma 3.1.Applying this lemma for P being the Scheepers property, we conclude thatthere exists a Scheepers semifilter F such that F + ⊂ F . For every n ∈ ω let us denote by O n the open subset { x ⊂ ω : n ∈ x } of P ( ω ) and notethat each x ∈ F belongs to infinitely many members of U = { O n : n ∈ ω } .Applying [21, Theorem 21] (namely the implication (1) → (2) there) weconclude that there exists an increasing number sequence ( n k ) k ∈ ω such that n = 0 and (cid:8) { [ O n : n ∈ [ n k , n k +1 } : k ∈ ω (cid:9) is an ω -cover of F . The latter means that for any family { x , . . . , x l } ⊂ F there exist infinitely many k ∈ ω such that x i ∩ [ n k , n k +1 ) = ∅ for all i ≤ l .Let us define φ : ω → ω by letting φ − ( k ) = [ n k , n k +1 ) for all k and set S = { s ⊂ ω : φ − ( s ) ∈ F } = { φ ( x ) : x ∈ F } . Then S is a Scheepers semifilter being a continuous image of F . Claim 3.2. S is centered.Proof. Given any s , . . . , s l ∈ S , set x i = φ − ( s i ) and take k ∈ ω such that x i ∩ [ n k , n k +1 ) = x i ∩ φ − ( k ) = ∅ for all i ≤ l . There are infinitely many such k ’s, and each of them is an element of s i for all i because s i = φ ( x i ). (cid:3) Claim 3.3. S + ⊂ S .Proof. Take any x ∈ S + and set y = φ − ( x ). Then y ∈ F + : given u ∈ F ,note that φ ( u ) ∈ S , and hence | φ ( u ) ∩ x | = ω, which implies that | u ∩ y | = ω and thus y meets all elements of F . Since F + ⊂ F , we have that y ∈ F ,and consequently x = φ ( y ) ∈ S . (cid:3) Let us fix now s , . . . , s i ∈ S and take arbitrary s ∈ S . Claim 3.2implies that | s ∩ T i ≤ l s i | = ω , hence T i ≤ l s i ∈ S + , and therefore T i ≤ l s i ∈ S by Claim 3.3. Thus S is a filter, and consequently it is an ultrafilter byClaim 3.3. This completes our proof. ✷ We call a semifilter F a P -semifilter if for every sequence ( F n ) n ∈ ω ∈ F ω there exists a sequence ( A n ) n ∈ ω such that A n ∈ [ F n ] <ω and S n ∈ ω A n ∈ F .Note that if F is a filter then we get a standard definition of a P -filter. P -filters which are ultrafilters are nothing else but P -points.Recall that for every n ∈ ω we denote by O n the clopen subset { x ⊂ ω : n ∈ x } of P ( ω ). The following fact is straightforward. Observation 3.4.
Let A ⊂ ω and F be a semifilter. Then { O n : n ∈ A } covers F + iff A ∈ F . Consequently, if F + is Menger, then F is a P -semifilter. ENGER REMAINDERS OF TOPOLOGICAL GROUPS 9
Proof of Corollary 1.5.
It is known [10] that under d = c there existsan ultrafilter F on ω such that the Mathias forcing M ( F ) does not adddominating reals, see [10] for corresponding definitions. Applying [11, The-orem 1] we conclude that F is Menger when considered with the topologyinherited from P ( ω ). By [11, Claim 5.5] we have that all finite powers of F are Menger, and hence F is Scheepers by [17, Theorem 3.9].Now suppose that F is a Menger ultrafilter. By the maximality of F wehave F = F + . Now it suffices to apply Observation 3.4. ✷ Menger remainders
This section is devoted to the proof of Theorem 1.6 which is divided intoa sequence of lemmata. In the proof of the next lemma we shall need thefollowing game of length ω on a topological space X : In the n th move player I chooses an open cover U n of X , and player II responds by choosing a finite V n ⊂ U n . Player II wins the game if S n ∈ ω S V n = X . Otherwise, player I wins. We shall call this game the Menger game on X . It is well-known that X is Menger if and only if player I has no winning strategy in the Mengergame on X , see [16] or [22, Theorem 13].Formally, a strategy for player I is a map § : τ <ω → O ( X ), where τ isthe topology of X and O ( X ) is the family of all open covers of X . Thestrategy § is winning if S n ∈ ω U n = X for any sequence ( U n ) n ∈ ω ∈ τ ω suchthat U n is a union of a finite subset of § ( U , . . . , U n − ) for all n ∈ ω . Lemma 4.1.
Suppose that F is a Menger semifilter. Then for every se-quence h B i : i ∈ ω i ∈ ( F + ) ω and increasing h ∈ ω ω there exists increasing δ ∈ ω ω such that [ i ∈ ω B i ∩ [ h (2 δ ( i )) , h (2 δ ( i + 1))) ∈ F + . Proof.
For every n ∈ ω let us denote by O n the subset { x ⊂ ω : n ∈ x } of P ( ω ) and note that O n is clopen. It is easy to see that for B ⊂ ω thecollection U B := { O n : n ∈ B } is an open cover of F if and only if B ∈ F + .Set δ (0) = 0 and consider the following strategy for player I in the Mengergame on F : In the 0th move he chooses U B \ h (0) = U B \ h ( δ (0)) . Supposethat for some i ∈ ω we have already defined δ ( i ). Then player I chooses U B i \ h (2 δ ( i )) . If player II responds by choosing V i ∈ [ U B i \ h (2 δ ( i )) ] <ω , then wedefine δ ( i + 1) to be so that V i ⊂ { O n : n ∈ [ h (2 δ ( i )) , h (2 δ ( i + 1))) ∩ B i } ,and the next move of player I is U B i +1 \ h (2 δ ( i +1)) .The strategy for player I we described above is not winning, so thereexists a run in the Menger game in which he uses this strategy and looses.Let δ be the function defined in the course of this run. It follows that S {V i : i ∈ ω } ⊃ F , where the V i s are the moves of player II , and hence [ i ∈ ω B i ∩ [ h (2 δ ( i )) , h (2 δ ( i + 1))) ∈ F + because V i ⊂ { O n : n ∈ [ h (2 δ ( i )) , h (2 δ ( i + 1))) ∩ B i } . (cid:3) For a semifilter F we denote by P F the poset consisting of all partial maps p from ω × ω to 2 such that for every n ∈ ω the domain of p n : k p ( n, k ) isan element of ∼ F . If, moreover, we assume that and dom( p n ) ⊂ dom( p n +1 )for all n , the corresponding poset will be denoted by P ∗F . A condition q isstronger than p (in this case we write q ≤ p ) if p ⊂ q . For filters F theposet P ∗F is obviously dense in P F , and the latter is proper and ω ω -boundingif F is a non-meager P -filter [23, Fact VI.4.3, Lemma VI.4.4]. In light ofObservation 3.4, the following lemma may be thought of as a topologicalcounterpart of [23, Fact VI.4.3, Lemma VI.4.4]. Lemma 4.2. If F + is a Menger semifilter, then both P F and P ∗F are properand ω ω -bounding.Proof. We shall present the proof for P F . The one for the poset P ∗F iscompletely analogous.To prove the properness let us fix a countable elementary submodel M ∋ P F of H ( θ ) for θ big enough, a condition p ∈ P F ∩ M , and list all open densesubsets of P F which are elements of M as { D i : i ∈ ω } . Let us denote by τ the collection of all open subsets of P ( ω ). For every s ∈ [ ω ] <ω we shalldenote by O s the set { x ⊂ ω : x ∩ s = ∅} . O s is clearly a clopen subset of P ( ω ).In what follows we shall define a strategy § : τ <ω → O ( F + ) of player I in the Menger game on F + as well as a map § : τ <ω ∩ M → P F ∩ M . Set p = p , § ( ∅ ) = p , and § ( ∅ ) = { O s : ∃ l ∈ ω [ s = ( ω \ dom( p )) ∩ l ] } . Now suppose that for some n ∈ ω and all sequences ( U k ) k Suppose that F = F + is a semifilter with Menger square. Let x be P ∗F -generic, Q ∈ V [ x ] be an ω ω -bounding poset, and H be a Q -genericover V [ x ] . Then in V [ x ∗ H ] there is no semifilter G = G + containing F such that G is Menger.Proof. Throughout the proof we shall identify x with ∪ x : ω × ω → G exists. Set x j ( n ) = x ( j, n ). In V [ x ∗ H ], the following 2 cases are possible.a). For every m ∈ ω there exists k > m such that S j ∈ [ m,k ) [ x j = x m ] ∈ G .Then we can inductively construct an increasing sequence h m k : k ∈ ω i suchthat(1) [ j ∈ [ m k ,m k +1 ) [ x j = x m k ] ∈ G for all k. Since P ∗F ∗ Q is ω ω -bounding, we may additionally assume that this sequenceis in V .b). There exists m such that S j ∈ [ m,k ) [ x j = x m ] ∈∼ G for all k > m .This means that T j ∈ [ m,k ) [ x j = x m ] ∈ G for all k > m . Then[ x i = x i +1 ] ⊃ [ x i = x m ] ∩ [ x i +1 = x m ] ⊃ \ j ∈ [ m,i +2) [ x j = x m ] ∈ G for all i > m . Thus the sequence m k = m + 1 + 2 k satisfies (1), and hencethere always exists a sequence h m k : k ∈ ω i ∈ V satisfying (1). Set A k = S j ∈ [ m k ,m k +1 ) [ x j = x m k ] ∈ G and U k = { U kn : n ∈ ω } , where U kn = {h X, Y i ∈ P ( ω ) : ∀ i ≤ k (cid:0) ( X ∩ A i ∩ [ k, n ) = ∅ ) ∧ ( Y ∩ A i ∩ [ k, n ) = ∅ ) (cid:1) } . Since A k ∈ G = G + for all k , U k is easily seen to be an open cover of G . TheMenger property of G yields a strictly increasing f ∈ ω ω ∩ V [ x ∗ H ] such that { U kf ( k ) : k ∈ ω } covers G . Since P ∗F ∗ Q is ω ω -bounding, we could additionallyassume that f ∈ V . Set h (0) = f (0) + 1 and h ( l + 1) = f ( h ( l )) + 1 for all l . Note that U kn = ( W kn ) , where W kn = { X ∈ P ( ω ) : ∀ i ≤ k ( X ∩ A i ∩ [ k, n ) = ∅ ) } . Therefore there exists ǫ ∈ O ǫ := [ { W kf ( k ) : k ∈ [ l ∈ ω [ h (2 l + ǫ ) , h (2 l + ǫ + 1)) } ⊃ G :If there were X ǫ ∈ G \ O ǫ for all ǫ ∈ 2, then h X , X i could not be anelement of U kf ( k ) for any k thus contradicting the choice of f . Without lossof generality ǫ = 0 is as above. Claim 4.4. Let δ ∈ ω ω be strictly increasing. Then A δ := [ i ∈ ω A i ∩ [ h (2 δ ( i )) , h (2 δ ( i + 1))) ∈ G . Proof. Given any X ∈ G , find l ∈ ω and k ∈ [ h (2 l ) , h (2 l + 1)) such that X ∈ W kf ( k ) . Let i ∈ ω be such that l ∈ [ δ ( i ) , δ ( i + 1)). Note that i ≤ l ≤ k ,hence X ∈ W kf ( k ) implies X ∩ A i ∩ [ k, f ( k )) = ∅ . It follows that[ k, f ( k )) ⊂ [ h (2 l ) , f ( h (2 l + 1))) ⊂ [ h (2 l ) , h (2 l + 2)) ⊂ [ h (2 δ ( i )) , h (2 δ ( i + 1))) , consequently X ∩ A i ∩ [ h (2 δ ( i )) , h (2 δ ( i + 1))) = ∅ , which implies S i ∈ ω A i ∩ [ h (2 δ ( i )) , h (2 δ ( i + 1))) ∈ G + . (cid:3) Let us fix any p ∈ P ∗F and set B i = ω \ supp( p m i +1 ) ∈ F + . ByLemma 4.1 used in V there exists an increasing δ such that B := S i ∈ ω B i ∩ [ h (2 δ ( i )) , h (2 δ ( i + 1))) ∈ F + = F . For every m ∈ ω find i such that m ∈ [ m i , m i +1 ) and set q m = p m ∪ (cid:0) B i ∩ [ h (2 δ ( i )) , h (2 δ ( i + 1))) × { } (cid:1) if m = m i and q m = p m ∪ (cid:0) B i ∩ [ h (2 δ ( i )) , h (2 δ ( i + 1))) × { } (cid:1) otherwise. This q obviously forces (i.e., any condition in P ∗F ∗ Q whose firstcoordinate is q forces) that B i ∩ ˙ A i ∩ [ h (2 δ ( i )) , h (2 δ ( i + 1))) = ∅ for all i ,and hence it also forces B ∩ ˙ A δ = ∅ . Thus the set of those q ∈ P ∗F whichforce B ∩ ˙ A δ = ∅ is dense, which means that B ∩ A δ = ∅ (here A δ = ˙ A δG ∗ H ).However, B ∈ F ⊂ G by the choice of δ and A δ ∈ G by Claim 4.4, andtherefore B ∩ A δ = ∅ contradicts G = G + . This contradiction completes ourproof. (cid:3) ENGER REMAINDERS OF TOPOLOGICAL GROUPS 13 Proof of Theorem 1.6. Suppose that there exists a topological group G anda compactification bG , such that ( bG \ G ) is Menger but not σ -compact.Then in the same way as at the beginning of the proof of Theorem 1.3 weconclude that G is feathered. By Lemma 2.1 we may assume without loss ofgenerality that G satisfies the premises of Lemma 3.1. Applying this lemmafor P being the property of having the Menger square we get a semifilter F = F + such that F is Menger. Thus the theorem will be proved as soonas we construct a model of ZFC in which there are no semifilters F = F + with Menger square.To this end let us assume that GCH holds in V and consider a function B : ω → H ( ω ), the family of all sets whose transitive closure has size < ω , such that for each x ∈ H ( ω ) the family { α : B ( α ) = x } is ω -stationary. Let h P α , ˙ Q β : β < α ≤ ω i be the following iteration with atmost countable supports: If B ( α ) is a P α -name for P ∗ ˙ F for some semifilter˙ F such that (cid:13) P α “ ˙ F = ˙ F + and ˙ F is Menger”, then ˙ Q α = P ∗ ˙ F . Otherwisewe let ˙ Q α to be a P α -name for the trivial forcing. Then P ω is ω ω -boundingforcing notion with ω -c.c. being a countable support iteration of length ω of proper ω ω -bounding posets of size ω over a model of CH.Let G be a P ω -generic over V and suppose that F ∈ V [ G ] is a semifiltersuch that F = F + and F is Menger. Then the set (cid:8) α : F α := ( F ∩ V [ G ∩ P α ]) ∈ V [ G ∩ P α ] , F α = F + α and F α is Menger in V [ G ∩ P α ] (cid:9) contains an ω -club subset of ω , and hence for one of these α we have that ˙ Q α = P ∗ ˙ F α ,where ˙ F α is a P α -name such that ˙ F G ∩ P α α = F ∩ V [ G ∩ P α ]. Now, a directapplication of Lemmata 4.3 and 4.2 implies that F α ⊂ F cannot be enlargedto any semifilter U ∈ V [ G ] such that U is Menger and U + = U , whichcontradicts our choice of F . ✷ Let us note that in the proof of Theorem 1.6 above we have also proventhe following Theorem 4.5. It is consistent with ZFC that there are no semifilters F such that F = F + and F is Menger. On a possible dichotomy for the Menger property Our first attempt to find a counterpart of the Menger property is basedon its game characterization we have exploited in Section 4. As the Mengergame produces a strengthening of the Lindel¨of property, we should considera game which produces a strengthening of the Baire property.There is an obvious candidate for this purpose: the Banach-Mazur game,see for instance [18] for more information. This game BM( X ) is played onthe space X in ω -many innings between two players α and β as follows. β makes the first move by choosing a non-empty open set U and α respondsby taking a non-empty open set V ⊆ U . In general, at the n-th inning β chooses a non-empty open set U n ⊆ V n − and α responds by takinga non-empty open set V n ⊆ U n . The rule is that α wins if and only if T { V n : n < ω } 6 = ∅ . The relationship of the Banach-Mazur game withBaire spaces is given by the following [18, Theorem 8.11]. Theorem 5.1. A space X is Baire if and only if player β does not have awinning strategy in BM ( X ) . Consequently, if α has a winning strategy, then the space is Baire. Definition. A space X is weakly α -favorable if player α has a winningstrategy in the Banach-Mazur game. X is said to be α -favorable if player α has a winning tactic, i. e. a winning strategy depending only on the lastmove of β . ✷ Every pseudocompact space is α -favorable: player α has an easy winningtactic by choosing for any U n a non-empty open set V n such that V n ⊆ U n . Of course, every weakly α -favorable space is Baire. Moreover, thefollowing observation shows that being weakly α -favorable often contradictsthe Menger property. Observation 5.2. No nowhere locally compact weakly α -favorable subset X of the real line is Menger.Proof. Since X is nowhere locally compact, we may assume that X ⊂ R \ Q ,and the latter we shall identify with ω ω . By [18, Theorem 8.17(1)] X ⊃ Y for some dense G δ subset Y of ω ω . By the Baire category theorem Y cannotbe contained in a σ -compact subspace of ω ω , and hence it contains a copy Z of ω ω which is closed in ω ω according to [18, Corollary 21.23]. Therefore Z is a closed in X copy of ω ω , which implies that X is not Menger as theMenger property is inherited by closed subspaces. (cid:3) Therefore, weak α -favorability seems to be a good candidate to be thecounterpart of the Menger property. However, this is not the case by The-orem 5.6 below. Let us recall that a set S ⊂ R is a Bernstein set providedthat both S and R \ S meet every closed uncountable subset of R . The nexttwo lemmas seem to be known, but we were not able to find them in theliterature. That is why we present their proofs. Lemma 5.3. There is a subgroup G of the real line R containing the ratio-nals which is a Bernstein set.Proof. Let { C α : α < c } be the collection of all closed uncountable subsets of R . Here, we will consider R as a Q -vector space. Choose a point x ∈ C anddenote by G the vector subspace of R generated by { , x } . Obviously, wehave | G | = ω . Then, pick a point y ∈ C \ G . We proceed by transfiniteinduction, by assuming to have already constructed a non decreasing familyof vector subspaces { G β : β < α } of R satisfying | G β | ≤ | β | + ω for each β and points x β , y β ∈ C β in such a way that x β ∈ G β and { y β : β <α } ∩ S { G β : β < α } = ∅ . The set H α = S { G β : β < α } has cardinality notexceeding | α | + ω and therefore even the vector subspace K α generated bythe set H α ∪{ y β : β < α } has cardinality less than c . So we may pick a point x α ∈ C α \ K α . Then, let G α be the vector subspace generated by H α ∪ { x α } and finally pick a point y α ∈ C α \ G α . It is clear that | G α | ≤ | α | + ω . Tocomplete the induction, we need to show y β / ∈ G α for each β < α . Indeed,if we had y β ∈ G α for some β , then y β = z + qx α , where z ∈ H α and ENGER REMAINDERS OF TOPOLOGICAL GROUPS 15 q ∈ Q \ { } . But, this would imply x α = q − z − q − z ∈ K α , in contrastwith the way x α was chosen.Now, we let G = S { G α : α < c } . It is clear that G is a Q -vectorsubspace, and hence a subgroup, of R which is also a Bernstein set. (cid:3) Lemma 5.4. A Bernstein set X ⊂ ω ω does not have the Menger property.Proof. For any f ∈ ω ω there exists some g ∈ X such that f ( n ) < g ( n )for each n ∈ ω . This comes from the fact that X must meet the Cantorset Q n<ω { f ( n ) + 1 , f ( n ) + 2 } . To finish, recall that a dominating subsetof ω ω is never Menger. Indeed, for any n < ω let π n : ω ω → ω be theprojection onto the n-th factor and put U n = { π − n ( k ) ∩ X : k ∈ ω } . Each U n is an open cover of X . For any choice of a finite set V n ⊆ U n , we maydefine a function g : ω → ω by letting g ( n ) = max π n ( S V n ), if V n = ∅ , and g ( n ) = 0 otherwise. Since X is dominating, there is some f ∈ X such that g ( n ) < f ( n ) for each n . Clearly, f / ∈ S { S V n : n < ω } and so X is notMenger. (cid:3) Lemma 5.5. A Bernstein set X ⊆ R is not weakly α -favorable.Proof. By [18, Theorem 8.17(1)] any weakly α -favorable subspace of R iscomeager, while no Bernstein set can be comeager because any comeagersubspace of R contains homeomorphic copies of the Cantor set. (cid:3) These three lemmas imply: Theorem 5.6. There exists a topological group G and its compactification bG such that the remainder bG \ G is neither Menger nor weakly α -favorable.Proof. Recall that the set of irrationals R \ Q is homeomorphic to ω ω . Let G be such as in Lemma 5.3. By Lemma 5.4 R \ G ⊆ ω ω is not Menger, andby Lemma 5.5 R \ G is not weakly α -favorable. Now, it suffices to take as bG the compactification of R obtained by adding two end-points. (cid:3) Theorem 5.6 implies that the counterpart of the Menger property shouldbe in between of weakly α -favorable and Baire.6. Miscellanea A very important example of a topological group is C p ( X ), the subspaceof R X with the Tychonoff product topology consisting of all continuous func-tions. We expect that the remainder of C p ( X ) cannot distinguish betweenbeing Menger and σ -compact, but we cannot prove this. Question 6.1. Is it true that a remainder of C p ( X ) is Menger if and onlyif it is σ -compact?Below we present some results giving a partial solution of Question 6.1. Proposition 6.2. Let Z be a compactification of C p ( X ) . If Z \ C p ( X ) isMenger, then C p ( X ) is first countable and hereditarily Baire. Proof. Since Menger spaces are Lindel¨of, by Henriksen-Isbell’s theorem [14], C p ( X ) is of countable type, and therefore it contains a compact subgroupwith countable outer base according to [3, Lemma 4.3.10]. It is easy tosee that there is no compact subgroup of C p ( X ) except for { } : for any f ∈ C p ( X ) \ { } , the set { nf : n ∈ ω } is not contained in any compact K ⊂ C p ( X ) because { nf ( x ) : n ∈ ω } is unbounded in R if f ( x ) = 0.Therefore C p ( X ) is first-countable, and hence X is countable.Since Z \ C p ( X ) is Menger, it follows that C p ( X ) contains no closedcopy of Q . Now, a theorem of Debs [12] implies that C p ( X ) is hereditarilyBaire. (cid:3) The following fact together with Proposition 6.2 gives the positive answerto Question 6.1 for spaces containing non-trivial convergent sequences. Observation 6.3. If X contains a non-trivial convergent sequence then C p ( X ) is not Baire.Proof. There is nice characterization of the Bairness for spaces of the form C p ( X ) due to Tkachuk. However, we shall present here a direct elementaryproof. Suppose that ( x n ) n ∈ ω is an injective sequence converging to x . Set F n = { f ∈ C p ( X ) : ∀ m ≥ n ( | f ( x ) − f ( x m ) | ≤ } . It is easy to check that each F n is closed nowhere dense in C p ( X ) and C p ( X ) = S n ∈ ω F n . (cid:3) By a theorem of Lutzer (see Problem 265 in [24]), C p ( X ) is ˇCech-complete if and only if X is countable and discrete. So to answer Ques-tion 6.1 in the affirmative we need to show that C p ( X ) has a Menger re-mainder only if X is countable and discrete.Note that in the proof of the “if” part of Theorem 1.4 the compactifi-cation was a topological group itself, namely ( P ( ω ) , ∆). We do not knowwhether complements to Menger subspaces in other Polish groups (e.g., R )may consistently be subgroups. The next proposition imposes some restric-tions. Proposition 6.4. Let G be an analytic topological group and M be a non-empty Menger subspace of G . If G \ M is a subgroup of G , then G is σ -compact and M contains a topological copy of P ( ω ) .Proof. Suppose that H = G \ M is a subgroup of G and fix g ∈ M . Then H ⊂ g − ∗ M , where ∗ is the underlying operation on G . Therefore G = M ∪ g − ∗ M is Menger, and hence it is σ -compact, see [1].Now suppose that M contains no topological copy of P ( ω ) and let X ⊂ G be homeomorphic to P ( ω ). If X ⊂ H then g ∗ X ⊂ g ∗ H ⊂ M which isimpossible by our assumption above. Thus X ∩ M = ∅ . Since M containsno copy of P ( ω ), X \ M is dense in X , and hence there exists a countabledense subset Q of X disjoint from M . Then X ∩ M = ( X \ Q ) ∩ M is aclosed subset of M . Note that ( X \ Q ) is a copy of ω ω and M ∩ ( X \ Q ) isa Bernstein set in ( X \ Q ). To finish, it suffices to apply Lemma 5.4. (cid:3) ENGER REMAINDERS OF TOPOLOGICAL GROUPS 17 The following statement shows that the classical Cantor-Bendixon induc-tive procedure does not have any variant allowing to separate a “nowhereperfect” core of a Menger space from its “ σ -compact part”. Proposition 6.5. There exists a Baire dense nowhere locally compact sub-group I of P ( ω ) with the Menger property such that for every σ -compact sub-space S of I there exists K ⊂ I homeomorphic to P ( ω ) such that K ∩ S = ∅ .Proof. It is well-known that there exists a non-meager Menger filter F on ω , see, e.g., the proof of Theorem 1 in [20]. Let I be the dual ideal of F . Then I is Menger, nowhere locally compact, and non-meager beinghomeomorphic to F . Also, I is a subgroup of P ( ω ), and hence it is Bairebecause each non-meager topological group is so. Note that I containscopies of P ( ω ): for every infinite I ∈ I the set P ( I ) ⊂ I is such a copy. Letus fix X ⊂ I homeomorphic to P ( ω ) and a σ -compact S ⊂ I . Then thereexists I ∈ I \ ( S + X ) because S + X is σ -compact and I is not. It followsthat K := { I } − X is a copy of P ( ω ) disjoint from S . (cid:3) Acknowledgement. The authors wish to thank Masami Sakai and BoazTsaban for many useful comments. 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