The Strength of Menger's Conjecture
aa r X i v : . [ m a t h . GN ] J un The Strength of Menger’s Conjecture
Franklin D. Tall , Stevo Todorcevic , Seçil Tokgöz June 30, 2020
Abstract
Menger conjectured that subsets of R with the Menger property mustbe σ -compact. While this is false when there is no restriction on thesubsets of R , for projective subsets it is known to follow from the Ax-iom of Projective Determinacy, which has considerable large cardinalconsistency strength. We note that in fact, Menger’s conjecture forprojective sets has consistency strength of only an inaccessible cardi-nal.
1. Introduction
In 1924, Menger [16] introduced a topological property for metric spaceswhich he referred to as “property E” . Hurewicz [10] reformulated propertyE as the following, nowadays called the
Menger property:
Definition 1.1.
A space X is Menger if whenever {U n } n ∈ ω is a sequence ofopen covers, there exist finite V n ⊆ U n , n ∈ ω , such that S n ∈ ω V n is a coverof X . Research supported by NSERC grant A-7354. Research supported by grants from CNRS and NSERC 455916. Research supported by TÜBİTAK grant 2219.2020 MSC. Primary 03E15, 03E35, 03E60, 54A35, 54D20, 54H05; Secondary 03E45.Keywords and phrases: Menger, Hurewicz, σ -compact, co-analytic, projective set of reals, L ( R ), Hurewicz Dichotomy. Hurewicz property is intermediate between Menger and σ -compact. Definition 1.2.
A space X is Hurewicz if for any sequence {U n } n ∈ ω of opencovers of X there are finite sets V n ⊆ U n such that { S V n : n ∈ ω } is a γ -cover of X , where an infinite open cover U is a γ -cover if for each x ∈ X the set { U ∈ U : x U } is finite. An equivalent definition (for completely regular spaces) is that a space X is Hurewicz if and only if for each Čech-complete space Z ⊇ X , there is a σ -compact space Y such that X ⊆ Y ⊆ Z [22], [3].There has recently been interest in the question of whether “definable”Menger spaces — and, more specifically, Menger sets of reals — are σ -compact. See e.g., [23, 24, 31]. Hurewicz [9] refuted under the ContinuumHypothesis Menger’s conjecture [16] that Menger subsets of R are σ -compact.Just et al. [12] refuted Hurewicz’s conjecture that Hurewicz sets of reals are σ -compact, and hence also refuted Menger’s conjecture in ZFC. A ZFC coun-terexample to Menger’s conjecture was earlier produced by Chaber and Pol[6] in an unpublished note. More natural examples were produced by Bar-toszyński and Shelah [4], and later Tsaban and Zdomskyy [33]. A convenientsource for examples differentiating these three properties is the survey paper[32].Hurewicz [9] proved that analytic Menger subsets of R are σ -compact;this was later extended to arbitrary Menger analytic spaces by Arhangel’ski˘ı[1]. Hurewicz [9] also proved this for completely metrizable spaces; this wasextended to Čech-complete spaces in [24]. That determinacy hypothesessuffice to imply more complicated “definable” Menger sets of reals (e.g. co-analytic ones) are σ -compact was first noticed in [19] and stated explicitly in[22]. See also [25] and [5].Determinacy hypotheses have considerable large cardinal strength, so itis of interest to compute the exact consistency strength of such propositionsas e.g. “every co-analytic (projective) Menger set of reals is σ -compact”.We shall consider three primary families of “definable” sets of reals: theco-analytic sets, the projective sets, and those sets of reals which are membersof L ( R ). The co-analytic sets are just the complements of analytic sets; theprojective sets are obtained by closing the Borel sets under complementationand continuous real-valued image. They are arranged in a hierarchy – seeKechris [14] for notation and properties. The co-analytic sets are also calledthe Π -sets. L ( R ) is the constructible closure of R . It is the smallest innermodel of ZF with R as a member. See e.g. Kanamori [13] or Moschovakis [20]2or its properties. It is frequently studied in its own right; for us, P ( R ) ∩ L ( R )– those sets of reals that are in L ( R ) – is a convenient large family of definablesets of reals that includes the projective sets and much more. For readersunfamiliar with definability, we point out that the process of constructing aBorel set, projective set, etc. can be encoded as a sequence of (sequences of. . . ) operations on rational intervals, and hence as a real number. Our mainresult is: Theorem 1.3.
The following are equiconsistent:a) there is an inaccessible cardinal,b) every Menger co-analytic set of reals is σ -compact,c) every Menger projective set of reals is σ -compact,d) every Menger set of reals in L ( R ) is σ -compact. As is common in descriptive set theory, we will use R or the Cantor setas convenient, since e.g. there is a co-analytic Menger non- σ -compact subsetof R if and only if there is one included in the Cantor set.
2. The Hurewicz Dichotomy
A classical phenomenon, the
Hurewicz Dichotomy , was first investigated byHurewicz [11] and later extended by Kechris, Louveau and Woodin [15]. Seee.g. Section 21 .F of [14]. Here is one version of the Hurewicz Dichotomy. Hurewicz Dichotomy (HD).
Let X be a Polish (separable completelymetrizable) space and A ⊆ X an analytic set. If A is not σ -compact, thenthere is a Cantor set K ⊆ X such that K ∩ A is dense in K and homeomor-phic to P , the space of irrationals, and K \ A is countable dense in K andhomeomorphic to Q , the space of rationals. Definition 2.1.
Let Γ be a subset of the power set of R . HD ( Γ ) is theassertion obtained from HD by replacing “analytic” by “ a member of Γ ” . Theorem 2.2 [11] . If Γ is a collection of subsets of R satisfying HD ( Γ ) asabove, then every Menger member of Γ is σ -compact. roof. Let A be a member of Γ. Suppose A is not σ -compact. By HD ( Γ ),there is a Cantor set K such that K ⊆ R and K ∩ A is homeomorphic to P . But K ∩ A is a closed subset of A and P is not Menger [10]; since thatproperty is closed-hereditary, A cannot be Menger. Remark.
The proof that an inaccessible suffices to prove the consistencywith ZFC of HD ( L ( R )) (of course we mean P ( R ) ∩ L ( R )) and hence thatMenger projective sets are σ -compact can actually be found in Di Prisco–Todorcevic [7]. This may not be obvious to the casual reader, since theauthors of [7] are interested in L ( R ) and other models not satisfying the Ax-iom of Choice. However the results about L ( R ) satisfying various principlessuch as the Hurewicz Dichotomy for all sets of reals can be interpreted asZFC results about sets of reals that happen to be in L ( R ). Remark.
Solovay [21] was the first to realize the usefulness of the model L ( R ) as computed in the forcing extension obtained by collapsing an inac-cessible cardinal to ω via finite conditions to problems of descriptive settheory such as for example the problem of Lebesgue measurability of projec-tive sets of reals. Feng’s paper [8] contains various interesting results aboutthis Solovay model L ( R ) , e.g. extensions of the fact that uncountable setsof reals which are in L ( R ) must include a perfect set. Solovay models arefurther explored in Di Prisco–Todorcevic [7] and Todorcevic [29]. Section 4plus point HD ( L ( R )) holds in such models.
3. The inaccessible is necessary
To prove Theorem 1.3, by the above Remarks it more than suffices to show:
Theorem 3.1. If ω L [ a ]1 = ω for some a ∈ R , then there is a co-analytic setof reals which is Hurewicz but not σ -compact. The reason is that then 1.3b) (and hence 1.3c) and 1.3d)) imply ω L [ a ]1 < ω for all a ∈ R and so ω L < ω . But then ω is inaccessible in L , so it’sconsistent there is an inaccessible. We shall rely on the following version ofa standard fact (see [13], p. 171). Lemma 3.2.
Assume ω L [ a ]1 = ω for some a ∈ R . Then ω ω ∩ L [ a ] orderedby the relation ≤ ∗ of eventual dominance has a co-analytic ω -scale, i.e., acofinal subset A which is well-ordered by ≤ ∗ in order type ω . roof of Theorem 3.1. Let A be the co-analytic set given by Lemma 3.2. Weknow that A is not σ -compact and in fact not Borel. This follows fromthe standard fact that a Borel well-founded relation on a Borel set of realshas countable rank (see [14, p. 239]). If A is Hurewicz, then the proof ofTheorem 3.1 is finished. Otherwise, by a theorem of Hurewicz [9], there isa continuous mapping f : A → ω ω whose range is unbounded in ( ω ω , ≤ ∗ ) . The map f extends to a continuous map on a G δ -superset of A . So there is aBorel map (also called a measurable map ) g : ω ω → ω ω such that g ↾ A = f .See Theorem 12 . b ∈ ω ω code both a andthe map g. Then ω ω ∩ L [ b ] is unbounded in ( ω ω , ≤ ∗ ). Applying Lemma 3.2again, we obtain a co-analytic ω -scale B in ( ω ω ∩ L [ b ] , ≤ ∗ ) . Since ω ω ∩ L [ b ]is unbounded in ( ω ω , ≤ ∗ ), that co-analytic ω -scale B is then a b -scale in ω ω ,i.e. an unbounded set { b α : α < b } such that the enumeration is increasingwith respect to ≤ ∗ . By [32, Theorem 3.3], B ∪ ω < ∞ is Hurewicz, so the proofof Theorem 3.1 is finished and hence so is the proof of Theorem 1.3.In [26], Tall and Zdomskyy show questions about whether Menger defin-able sets of reals are σ -compact are essentially equivalent to questions aboutwhether completely Baire definable sets of reals are Polish, where a space iscompletely Baire if each closed subspace satisfies the Baire Category Theo-rem. A result needed for their work (which was written after seeing an earlypreprint of this paper) is that: Theorem 3.3.
If it is consistent there is an inaccessible cardinal, it is con-sistent that every completely Baire projective set of reals is Polish.Proof.
We have stated in Section 2 that the consistency of an inaccessibleyields the consistency of
HD(projective) . From that it is easy to prove thedesired result—see the proof of 21.21 from 21.18 in [14].
4. A co-analytic gap theorem and the HurewiczDichotomy
We shall need the following which is a variant of Theorem 3 of [28] given in[30]. For a family of subsets of ω denoted by B , a subset Σ ⊆ [ ω ] <ω is calleda B-tree [28] if 5i) ∅ ∈
Σ, (ii) for every σ ∈ Σ, the set { i ∈ ω : σ ∪ { i } ∈ Σ } is infinite andincluded in an element of B . Theorem 4.1.
A co-analytic gap theorem ( CAG ) . Suppose ω L [ x ]1 < ω for all x ⊆ ω . Let A and B be two orthogonal families of subsets of ω closed downwards such that A is co-analytic and B is analytic or co-analytic.Then either A is countably generated in B ⊥ or there is a B -tree all of whosebranches are in A .Proof. For the convenience of the reader we sketch the argument from [30].The proof is, in fact, a straightforward variation of the proof of Theorem3 of [28]. We start the proof by fixing a real x and two downwards closedsubtrees T and S of [ ω ] <ω ⊗ [ ω ] <ω ordered by end-extension and belongingto L [ x ] for some x ⊆ ω such that A = p [ T ] and B = p [ S ]. (See e.g. [18,p.86].) For a subtree U of T and t = ( t , t ) ∈ U, let U ( t ) denote the subtreeof U consisting of all nodes of U comparable to t. For a downwards closedsubtree U of T, let ∂U = { t ∈ U : [ p [ U ( t )] B ⊥ } . Note that by absoluteness, if U belongs to L [ x ] so does ∂U. Let T (0) = T , T ( α +1) = ∂T α and T λ = T α<λ T α for limit ordinal λ. Let β be the minimal ordinal α with the property that T ( α +1) = T α . If T β = ∅ ,then working as in the proof of Theorem 3 of [28], we get a B -tree all ofwhose branches are in A. If T β = ∅ , then for every a ∈ A there are α < β and t ∈ T ( α ) \ T α +1 such that a ∈ p [ T ( α ) ( t )] and therefore a ⊆ b ( α, t ) = S p [ T ( α ) ( t )] . We have already noted that the trees of the form T ( α ) ( t ) belongto L [ x ] , so we have that the sets b ( α, t ) are also elements of L [ x ] . Since allthese sets are in B ⊥ , we have that A is generated by B ⊥ ∩ L [ x ] . Since byour assumption ω L [ x ]1 < ω , this set is countable, so we conclude that A iscountably generated in B ⊥ . Remark.
In [29] it is shown that in the Solovay model the conclusion of 4 . A and B are definable from finite sets of reals and ordinals.Thus, in particular. the consistency of 1.3a) implies the consistency of 1.3d). Theorem 4.2. CAG implies HD ( Π ) . roof of Theorem 4.2. Suppose A is a co-analytic not σ -compact set. Weshall find a counterexample to the conclusion of the co-analytic gap theorem.First of all, we may assume A is a subset of the Cantor set 2 ω . Let ˆ A be thecollection of all infinite chains of the Cantor tree 2 <ω whose union belongsto A. Note that ˆ A is a co-analytic collection of infinite subsets of 2 <ω . Let B = ˆ A ⊥ . We shall show that both alternatives of CAG fail for the gap( ˆ
A, B ) . First of all note that ˆ A is not a countably generated ideal sinceotherwise A would be a σ -compact set. Since B ⊥ = ˆ A this shows that thefirst alternative of CAG fails for the gap ( ˆ A, B ) . So, we are left with thealternative that there is a B -tree Σ all of whose infinite branches are in ˆ A. Thus, Σ is a collection of finite subsets of 2 <ω such that ∅ ∈ Σ and if t ∈ Σ , then Σ( t ) = { σ ∈ <ω : t ∪ { σ }} is an infinite set belonging to B. To viewΣ as a tree we order 2 <ω in order type ω extending the partial ordering ofend-extension in some natural way; we assume that for every t ∈ Σ, every σ ∈ Σ( t ) is above every τ ∈ t. This allows us to define a tree ordering on Σby letting s ⊑ t if and only if s ⊆ t and every element of s is smaller thanevery element of t \ s in the ω -ordering of 2 <ω just fixed. Define a one-to-onemapping φ : 2 <ω → <ω as follows. The definition is by recursion on the ω -ordering. Let φ ( ∅ ) = ∅ . Suppose φ ( τ ) is defined. Let τ [1] be the largestinitial segment of τ with last digit 1; if such initial segment does not exist,put τ [1] = ∅ . Let φ ( τ ⌢
0) be the minimal available element of Σ( φ ( τ )) and let φ ( τ ⌢
1) be the minimal available element of Σ( τ ) . The following properties of φ : 2 <ω → <ω are easy to verify. If x ∈ ω has infinitely many 1’s, then the φ -image of the infinite chain c x = { x ↾ n : x ( n ) = 1 } is an infinite chain ofΣ and therefore an element of ˆ A. On the other hand, if x ∈ ω is eventually0 then the φ -image of the chain c x = { x ↾ n : x ( n ) = 1 } ∪ { x ↾ n : n > n x } , where n x is the maximal integer where x has a digit 1, belongs to the family B. Note that x φ [ c x ] is a continuous map from 2 ω into the power-set of2 <ω , viewed as 2 <ω . So if we let P = { φ [ c x ] : x ∈ ω } , we get a copy ofthe Cantor set inside 2 <ω such that P \ ˆ A is a countable dense subset of P . It follows that the complement of ˆ A in 2 <ω includes a closed copy of therationals, so it can’t be analytic, a contradiction. This finishes the proof.We have the following consequence. Corollary 4.3.
The following are equivalent:a) ω L [ a ]1 < ω for all a ∈ R , ) co-analytic Menger subsets of R are σ -compact,c) co-analytic Hurewicz subsets of R are σ -compact. Remark.
An early version of this paper (The Open Graph Axiom andMenger’s Conjecture, Arxiv.org) claimed that the Open Graph Axiom (for-merly known as the Open Coloring Axiom of [27], but renamed to avoidconfusion with the identically named axiom of [2]) for co-analytic sets im-plied co-analytic (projective) Menger sets of reals are σ -compact. The proofwas flawed, but the result is true because of Theorems 4.1 and 4.2 and thefollowing result. Theorem 4.4 [8] . The following are equivalent:1. ω L [ a ]1 < ω for all a ∈ R ,2. ω is inaccessible in L [ a ] , any a ∈ R ,3. OGA ∗ ( Π ) (the co-analytic axiom referred to above). [26] was in press as we were revising this paper; the reference to OGA ∗ there is not incorrect but is irrelevant. Remark.
There is already a rich body results about the inner model L ( R )especially when it is a Solovay model (i.e., computed in the forcing exten-sions of the Levy collapse of an inaccessible cardinal to ω ) and thereforefails to satisfy the Axiom of Choice but rather satisfies strong descriptiveset-theoretic regularity properties for all sets of reals. In [7] , Di Priscoand Todorcevic compare the Solovay model L ( R ) and its (forcing) extension L ( R )[ U ], which is obtained from that L ( R ) by adjoining a selective ultra-filter U . They note that OGA ∗ ( P ( R )) holds in the latter model, but that HD ( P ( R )) does not. Thus one has to be careful about asserting implicationsfrom various forms of OGA ∗ to corresponding forms of HD . The counterex-ample to HD is defined from the generic ultrafilter U and, therefore, themodel L ( R )[ U ] satisfies OGA ∗ ( L ( R )[ U ]) but fails to satisfy HD ( L ( R )[ U ]).Here is another version of Theorem 3.3 : Theorem 4.5. If ω L [ a ]1 < ω for all a ∈ R , then every analytic completelyBaire subset of R is a G δ . Theorem 4.5 is used in [26], replacing “a G δ ” by “Polish”, which is obvi-ously equivalent. 8 emma 4.6 [11],[17] . B ⊆ R is completely Baire if and only if B does notinclude a closed copy of Q .Proof of Theorem 4.5. Suppose B is completely Baire, analytic, but not a G δ . Then by HD ( Π ) , since R \ B is co-analytic and not σ -compact, thereis a compact K with K ∩ B homeomorphic to Q . But K ∩ B is closed in B ,contradicting B being completely Baire.We can now add an additional clause to Corollary 4.3:e) every analytic completely Baire subset of R is a G δ .The reason is that in [26] it is established that if analytic completely Bairesubsets of R are Polish, then co-analytic Menger subsets of R are σ -compact. Problem 1.
Is there a model in which every co-analytic Menger set of realsis σ -compact, but there is a projective Menger set of reals which is not σ -compact? Problem 2.
If there is a co-analytic Menger subset of R which is not σ -compact, is there one which is not Hurewicz? In conclusion, we thank the referee for pointing out several inaccuraciesin the previous version of this note.
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Scales, fields, and a problem of Hurewicz ,J. Eur. Math. Soc. (2008), 837–866.Franklin D. Tall, Department of Mathematics, University of Toronto,Toronto, Ontario M5S 2E4, CANADA e-mail address: [email protected] Todorcevic, Department of Mathematics, University of Toronto,Toronto, Ontario M5S 2E4, CANADA e-mail address: [email protected]çil Tokgöz, Department of Mathematics, Hacettepe University,Beytepe, 06800, Ankara, TURKEY e-mail address:e-mail address: