aa r X i v : . [ m a t h . L O ] J a n COVERING VERSUS PARTITIONINGWITH POLISH SPACES
WILL BRIAN
Abstract.
Given a completely metrizable space X , let par ( X ) denotethe smallest possible size of a partition of X into Polish spaces, and cov ( X ) the smallest possible size of a covering of X with Polish spaces.Observe that cov ( X ) ≤ par ( X ) for every X , because every partition of X is also a covering.We prove it is consistent relative to a huge cardinal that the strict in-equality cov ( X ) < par ( X ) can hold for some completely metrizable space X . We also prove that using large cardinals is necessary for obtainingthis strict inequality, because if cov ( X ) < par ( X ) for any completelymetrizable X , then 0 † exists. Introduction
In this paper we examine coverings and partitions of large completelymetrizable spaces with small ones. By a “small” completely metrizable spacewe mean one with a countable basis, or (equivalently) a countable densesubset. This is the familiar class of Polish spaces. A “large” completelymetrizable space means anything else.For each completely metrizable space X , define cov ( X ) = min {|C| : C is a covering of X with Polish spaces } , par ( X ) = min {|P| : P is a partition of X into Polish spaces } . Observe that cov ( X ) and par ( X ) are well defined and both ≤ | X | , becausewe may cover/partition X with singletons. Also, cov ( X ) ≤ par ( X ) for every X , because every partition of X is also a covering of X . We are interestedin two basic questions: (1) Can we determine cov ( X ) or par ( X ) in terms ofmore familiar invariants? (2) Is it possible to have cov ( X ) < par ( X )?Another preliminary observation is that cov ( X ) = par ( X ) = | X | whenever | X | > c . This is because every Polish space has cardinality ≤ c , and thereforea collection F of Polish spaces covers at most |F | · c points.This gives an easy answer to both of our questions above when | X | > c .Consequently, we are mostly interested in covering and partitioning non-Polish spaces with cardinality ≤ c . If X is such a space, then ℵ ≤ cov ( X )(otherwise X would be separable) and par ( X ) ≤ c (because we may partition X into singletons). In this way, cov ( X ) and par ( X ) bear some resemblanceto cardinal characteristics of the continuum. Mathematics Subject Classification.
Primary: 54E35, 03E05 Secondary: 03E55.
Key words and phrases. covering, partitioning, metrizable spaces, large cardinals.
Recall that the weight of a topological space X , denoted wt ( X ), is theleast cardinality of a basis for X . The main results of this paper are: ◦ If X is any completely metrizable space with ℵ ≤ wt ( X ) < ℵ ω ,then cov ( X ) = par ( X ) = wt ( X ). ◦ It is consistent relative to a huge cardinal that cov ( X ) < par ( X ) for abroad class of completely metrizable spaces X with ℵ ω ≤ wt ( X ) < c .This can be achieved, for example, by adding > ℵ ω +1 Cohen reals toa model of
GCH + the Chang Conjecture ( ℵ ω +1 , ℵ ω ) ։ ( ℵ , ℵ ). ◦ Suppose that for every singular cardinal µ < c with cf( µ ) = ω ,a weak form of (cid:3) µ holds (namely (cid:3) ∗∗∗ ω ,µ ), and cf (cid:0) [ µ ] ℵ , ⊆ (cid:1) = µ + .Then cov ( X ) = par ( X ) for every completely metrizable space X , andthe value of cov ( X ) and par ( X ) is determined by basic topologicalproperties of X (see Theorem 4.8). Furthermore, the negation ofthese hypotheses implies 0 † exists; consequently, if cov ( X ) < par ( X )for any completely metrizable space X , then 0 † exists.These results are proved in Sections 2, 3, and 4, respectively.The second and third bullet points above answer a question raised severalyears ago by the author and Arnie Miller [2, Question 3.8]. At least, thequestion is answered to the satisfaction of the first author of [2].2. cov and par below ℵ ω For a set A and a cardinal µ < | A | , recall that [ A ] µ denotes the set of allsubsets of A with cardinality µ . A subset D of [ X ] µ is cofinal in (cid:0) [ A ] µ , ⊆ (cid:1) if every member of [ A ] µ is included in some member of D , and we definecf (cid:0) [ A ] µ , ⊆ (cid:1) = min (cid:8) |D| : D is cofinal in (cid:0) [ A ] µ , ⊆ (cid:1)(cid:9) . For example, cf (cid:0) [ ω ] ℵ , ⊆ (cid:1) = ℵ , because (identifying an ordinal with the setof its predecessors, as usual) D = { α : ω ≤ α < ω } is cofinal in (cid:0) [ ω ] ℵ , ⊆ (cid:1) .Given a topological space X , recall that a cellular family in X is a col-lection S of pairwise disjoint, nonempty open subsets of X . The cellularity of X , denoted c ( X ), is defined as c ( X ) = sup {|S| : S is a cellular family in X } . We will need the following two facts about cellularity and weight:
Lemma 2.1. If X is a metric space of weight κ , then(1) c ( X ) = κ .(2) X has a basis B of size κ such that every x ∈ X is contained in onlycountably many members of B .Proof. For the first assertion, see [6, Theorem 4.1.15]. For the second asser-tion, first recall that by the Bing Metrization Theorem, X has a basis B that is a countable union of cellular families. Every x ∈ X is contained inonly countably many members of B , and this remains true for any B ⊆ B .By [6, Theorem 1.1.15], X has a basis B ⊆ B with |B| = κ . (cid:3) OVERING VERSUS PARTITIONING WITH POLISH SPACES 3
Another fact used frequently throughout the paper is:
A subspace of acompletely metrizable space is completely metrizable if and only if it is a G δ (see [6, Theorems 4.3.23 and 4.3.24]). Theorem 2.2.
Let κ be an uncountable cardinal, and let X be any com-pletely metrizable space of weight κ . Then κ ≤ cov ( X ) ≤ cf (cid:0) [ κ ] ℵ , ⊆ (cid:1) . Furthermore, these bounds are attained: if κ is given the discrete topology,then both κ and κ ω are completely metrizable spaces of weight κ , cov ( κ ) = κ ,and cov ( κ ω ) = cf (cid:0) [ κ ] ℵ , ⊆ (cid:1) .Proof. Let κ > ℵ , and fix a completely metrizable space X of weight κ .Let C be a collection of Polish subspaces of X with |C| < κ . Because c ( X ) = κ (by Lemma 2.1(1)), there is a cellular family S in X with |C| < |S| .Each member of C has countable cellularity, and therefore each member of C meets only countably many members of S . Hence S C meets at most ℵ · |C| members of S . But C is uncountable (otherwise X would be a countableunion of separable subspaces, hence separable), so ℵ · |C| = |C| < |S| . Thusthere are U ∈ S with U ∩ S C = ∅ , which means that C does not cover X .Therefore cov ( X ) ≥ κ .Applying Lemma 2.1(2), let B be a basis for X with |B| = κ , such thatevery point of X is contained in only countably many members of B . Let D ⊆ [ B ] ℵ be cofinal in (cid:0) [ B ] ℵ , ⊆ (cid:1) , and for each A ∈ D define Y A = { x ∈ X : A ⊇ { U ∈ B : x ∈ U }} . It is straightforward to check that each Y A is a closed subspace of X , hencecompletely metrizable. Also, each of the Y A is second countable, with { Y A ∩ U : U ∈ A } as a basis. Therefore { Y A : A ∈ D} is a collection of Polishsubspaces of X . Also, { Y A : A ∈ D} covers X : if x ∈ X then { U ∈ B : x ∈ U } is countable by our choice of B , and so { U ∈ B : x ∈ U } ⊆ A for some A ∈ D ,by our choice of D . Hence cov ( X ) ≤ cf (cid:0) [ B ] ℵ , ⊆ (cid:1) = cf (cid:0) [ κ ] ℵ , ⊆ (cid:1) .This completes the proof of the first part of the theorem, and it remainsto prove the “furthermore” part. It is clear that both κ and κ ω (where κ is given the discrete topology) are completely metrizable spaces of weight κ , and that cov ( κ ) = κ . Also, it follows from what we already proved that cov ( κ ω ) ≤ cf (cid:0) [ κ ] ℵ , ⊆ (cid:1) . It remains to prove cov ( κ ω ) ≥ cf (cid:0) [ κ ] ℵ , ⊆ (cid:1) .Let C be a covering of κ ω with Polish spaces, and for each Y ∈ C define A Y = { x ( n ) : x ∈ Y and n ∈ ω } . Here, as usual, we are identifying the points of κ ω with functions ω → κ . Aseach Y ∈ C has countable cellularity, { x ( n ) : x ∈ Y } is countable for each n ; hence S n<ω { x ( n ) : x ∈ Y } = A Y ∈ [ κ ] ℵ for every Y ∈ C . If B ∈ [ κ ] ℵ ,then let x : ω → B be an enumeration of B . Because C covers κ ω , there issome Y ∈ C with x ∈ Y , and this implies B ⊆ A Y . Therefore { A Y : Y ∈ C} is cofinal in [ κ ] ℵ . Because C was an arbitrary covering of κ ω with Polishspaces, this implies cf (cid:0) [ κ ] ℵ , ⊆ (cid:1) ≤ cov ( κ ω ). (cid:3) WILL BRIAN
We observed above that cf (cid:0) [ ω ] ℵ , ⊆ (cid:1) = ℵ because { α : ω ≤ α < ω } iscofinal in (cid:0) [ ω ] ℵ , ⊆ (cid:1) . Similarly, cf (cid:0) [ ω n ] ℵ n − , ⊆ (cid:1) = ℵ n for all n >
0, because { α : ω n − ≤ α < ω n } is cofinal in (cid:0) [ ω n ] ℵ n − , ⊆ (cid:1) . By induction, this can beused to show cf (cid:0) [ ω n ] ℵ , ⊆ (cid:1) = ℵ n for all n >
0. Combining this equality withTheorem 2.2, we obtain the following:
Corollary 2.3.
Let X be a completely metrizable space and suppose that wt ( X ) = ℵ n for some n ∈ ω \ { } . Then cov ( X ) = ℵ n . More generally, for any cardinal κ with κ = cf (cid:0) [ κ ] ℵ , ⊆ (cid:1) , cov ( X ) = κ forevery completely metrizable space X of weight κ . This is fairly informative:if 0 † does not exist, then κ = cf (cid:0) [ κ ] ℵ , ⊆ (cid:1) for all κ with cf( κ ) > ω . Thisis discussed further in Section 4. On the other hand, if κ is uncountableand cf( κ ) = ω , then cf (cid:0) [ κ ] ℵ , ⊆ (cid:1) > κ . (This is proved by a simple diagonalargument. If cf (cid:0) [ κ ] ℵ , ⊆ (cid:1) ≤ κ , we would have a cofinal subset S n<ω D n of[ κ ] ℵ with |D n | < κ for all n , but then choosing α n ∈ κ \ S D n for all n , { α n : n ∈ ω } would not be contained in any member of any D n .)The situation for par is different. We shall see in the next section that par ( X ) > cf (cid:0) [ ω ω ] ℵ , ⊆ (cid:1) is consistent, relative to a huge cardinal, for cer-tain completely metrizable spaces X of weight ℵ ω . In particular, the upperbound for cov ( X ) in Theorem 2.2 does not necessarily apply to par ( X ).Nonetheless, we now show, via an inductive argument somewhat similar tothat preceding Corollary 2.3, that par ( X ) = wt ( X ) whenever wt ( X ) < ℵ ω . Theorem 2.4.
Let κ be a cardinal with uncountable cofinality, and let X be a completely metrizable space of weight κ . There is a size- κ partition of X into completely metrizable spaces of weight < κ .Proof. Using Lemma 2.1(2), let B = { U α : α < κ } be a basis for X such thateach x ∈ X is contained in only countably many members of B . For each α < κ , define X α = { x ∈ X : sup { ξ < κ : x ∈ U ξ } ≤ α } and Y α = X α \ S ξ<α X ξ . By our choice of B , { ξ < κ : x ∈ U ξ } is countable for each x ∈ X . Becausecf( κ ) > ω , this implies S α<κ X α = X . It follows that { Y α : α < κ } is apartition of X . Also, { X α ∩ U ξ : ξ ≤ α } is a basis for X α , and therefore wt ( Y α ) ≤ wt ( X α ) ≤ | α | < κ for each α < κ .To finish the proof, we must show that each of the Y α is completelymetrizable. It is not difficult to see that each of the X α is closed in X , andthat the X α are increasing, i.e., β < α implies X β ⊆ X α . If α = β + 1 is asuccessor ordinal, then Y α = X α \ X β (because the X α are increasing). Itfollows that Y α is a G δ subset of X , hence completely metrizable. If α isa limit ordinal with cf( α ) = ω , fix an increasing sequence h β n : n ∈ ω i withlimit α . Then Y α = X α \ S n ∈ ω X β n , so that once again Y α is a G δ in X ,hence completely metrizable. Finally, if α is a limit ordinal with cf( α ) > ω ,then Y α = ∅ , because by our choice of B , sup { ξ < κ : x ∈ U ξ } cannot haveuncountable cofinality, and so x ∈ X α implies x ∈ X ξ for some ξ < α . (cid:3) OVERING VERSUS PARTITIONING WITH POLISH SPACES 5
Theorem 2.5.
Let X be a completely metrizable space and suppose that wt ( X ) = ℵ n for some n ∈ ω \ { } . Then cov ( X ) = par ( X ) = ℵ n .Proof. That cov ( X ) = ℵ n is given by Corollary 2.3. That par ( X ) ≤ ℵ n follows from the previous theorem and a straightforward induction, and thereverse inequality is given in Theorem 2.2. (cid:3) Corollary 2.6.
Suppose that there is a completely metrizable space X with cov ( X ) < par ( X ) . Then c ≥ ℵ ω +1 .Proof. Suppose c < ℵ ω +1 . Then c < ℵ ω , because c cannot have countablecofinality. If wt ( X ) ≤ c , then cov ( X ) = par ( X ) by the previous theorem.If wt ( X ) > c , then | X | > c and (as pointed out in the introduction) thisimplies cov ( X ) = par ( X ) = | X | . (cid:3) Corollary 2.7.
Let X completely metrizable, with κ = wt ( X ) < ℵ ω . Ifthere is a continuous bijection X → [0 , , then there is a partition of [0 , into κ Borel sets.Proof.
Suppose f : κ ω → [0 ,
1] is a continuous bijection. By the previoustheorem, there is a partition P of κ ω into κ Polish spaces. By a theorem ofLusin and Suslin [15, Theorem 15.1], if B ⊆ [0 ,
1] is a continuous bijectiveimage of a Polish space, then B is Borel. Hence { f [ X ] : X ∈ P} is a partitionof [0 ,
1] into Borel sets. (cid:3)
Theorems 2.4 and 2.5 are reminiscent of Theorem 3.5 and Corollary 3.6in [2]. In fact, these results from [2] motivated much of the present paper.They state, respectively (giving all ordinals the discrete topology):(3 .
5) For each ordinal α , the space ω ωα +1 can be partitioned into ℵ α +1 copies of ω ωα .(3 .
6) For each n ∈ ω \ { } , ω ωn can be partitioned into ℵ n copies of theBaire space ω ω .We note in passing that (3 .
5) and (3 .
6) can be seen as special cases ofTheorems 2.4 and 2.5, respectively. Thus to some extent, the results ofthis section subsume these results from [2]. To deduce (3 .
5) and (3 .
6) fromTheorems 2.4 and 2.5, all one needs is the following lemma (whose proof weomit): If X is a zero-dimensional, completely metrizable space of weight ≤ κ ,then X × κ ω ≈ κ ω . Armed with this lemma, we see that if P is a partitionof ω ωα +1 as described in Theorem 2.4, then { Z × ω ωα : Z ∈ P} is a partitionof ω ωα +1 × ω ωα ≈ ω ωα +1 into ℵ α +1 copies of ω ωα . Similarly, if P is a partitionof ω ωn into ℵ n Polish spaces as in Theorem 2.5, then { Z × ω ω : Z ∈ P} is apartition of ω ωn × ω ω ≈ ω ωn into ℵ n copies of ω ω .3. A model for cov < par In this section, every ordinal, when referred to as a topological space, car-ries the discrete topology. We write ω ωω rather than ( ω ω ) ω for the countablepower of the size- ℵ ω discrete space ω ω . WILL BRIAN
The main result of this section is that it is consistent relative to a hugecardinal that cov ( ω ωω ) < par ( ω ωω ). The space ω ωω has weight ℵ ω , so this resultis optimal in some sense: by Theorem 2.5, completely metrizable spaces X of smaller weight must have cov ( X ) = par ( X ).Recall that cov( M ) denotes the smallest cardinality of a collection F ofmeager subsets of the Baire space ω ω with S F = ω ω . The use of ω ω in thisdefinition is inessential: if Y is any Polish space without isolated points,then cov( M ) is the smallest cardinality of a collection F of meager subsetsof Y with S F = Y . (For a proof that these two ways of defining cov( M )really are equivalent, see [9, Proposition 2].)Suppose X is a completely metrizable space, and P is a partition of X into completely metrizable subspaces. We say that P is skinny if for everyPolish Y ⊆ X , { Z ∈ P : Z ∩ Y = ∅} is countable. Lemma 3.1.
Suppose X is a completely metrizable space, and P is a par-tition of X into completely metrizable subspaces. If P is not skinny, then |P| ≥ cov( M ) .Proof. This follows almost immediately from a result of Fremlin and She-lah (which answered a question going back to Hausdorff) [9, Theorem 3]:
Any partition of a Polish space into uncountably many G δ sets is of size ≥ cov( M ) . Suppose X is a completely metrizable space, and P is a partition of X into completely metrizable subspaces. If P is not skinny, there is somePolish Y ⊆ X such that { Z ∈ P : Y ∩ Z = ∅} is uncountable. But then { Y ∩ Z : Z ∈ P} \ {∅} is an uncountable partition of Y into G δ sets, so itfollows from Fremlin and Shelah’s theorem that |P| ≥ cov( M ). (cid:3) Consider the following statement, abbreviated ( κ + , κ ) ։ ( µ + , µ ):For every model M for a countable language L that contains a unarypredicate A , if | M | = κ + and | A | = κ then there is an elementarysubmodel M ′ ≺ M such that | M ′ | = µ + and | M ′ ∩ A | = µ .The statement ( κ + , κ ) ։ ( µ + , µ ) is an instance of Chang’s conjecture . Toprove the main theorem of this section, we use the statement obtained bytaking κ = ℵ ω and µ = ℵ above. This instance of Chang’s conjecture,abbreviated ( ℵ ω +1 , ℵ ω ) ։ ( ℵ , ℵ ), is known as Chang’s conjecture for ℵ ω .The usual Chang conjecture, which is the assertion ( ℵ , ℵ ) ։ ( ℵ , ℵ ),is equiconsistent with the existence of an ω -Erd˝os cardinal. Chang’s con-jecture for ℵ ω requires even larger cardinals. GCH + ( ℵ ω +1 , ℵ ω ) ։ ( ℵ , ℵ )was first proved consistent relative to a hypothesis a little weaker than theexistence of a 2-huge cardinal in [18]. Recently this was improved to a hugecardinal in [7]. The precise consistency strength of ( ℵ ω +1 , ℵ ω ) ։ ( ℵ , ℵ )is an open problem, but significant large cardinal strength is known to beneeded. This is because ( ℵ ω +1 , ℵ ω ) ։ ( ℵ , ℵ ) implies the failure of (cid:3) ℵ ω (see[20], in particular Fact 4.2 and the remarks after it), and the failure of (cid:3) ℵ ω carries significant large cardinal strength (see [3]). OVERING VERSUS PARTITIONING WITH POLISH SPACES 7
Lemma 3.2. ( ℵ ω +1 , ℵ ω ) ։ ( ℵ , ℵ ) implies that no partition of ω ωω intoPolish spaces is skinny.Proof. Let P be a pairwise disjoint collection of Polish subspaces of ω ωω .By Theorem 2.2, cov ( ω ωω ) = cf (cid:0) [ ω ω ] ℵ , ⊆ (cid:1) , and a straightforward diagonalargument shows cf (cid:0) [ ω ω ] ℵ , ⊆ (cid:1) > ℵ ω . Hence |P| ≥ par ( ω ωω ) ≥ cov ( ω ωω ) ≥ℵ ω +1 . For each Y ∈ P , let A Y = { x ( n ) : x ∈ Y and n ∈ ω } . As in the proof of Theorem 2.2, A Y is countable for each Y ∈ P .Let ( M, ∈ ) be a model of (a sufficiently large fragment of) ZFC such that ω ω ⊆ M , P ∈ M , and | M | = | M ∩ P| = ℵ ω +1 . (Such a model can beobtained in the usual way, via the downward L¨owenheim-Skolem Theorem.)Let φ : M → M ∩ P be a bijection, and consider the model ( M, ∈ , φ, ω ω ) forthe 3-symbol language consisting of a binary relation, a unary function, anda unary predicate. Applying the Chang conjecture ( ℵ ω +1 , ℵ ω ) ։ ( ℵ , ℵ ),there exists some M ′ ⊆ M such that | M ′ | = ℵ , M ′ ∩ ω ω is countable, and( M ′ , ∈ , φ, ω ω ) ≺ ( M, ∈ , φ, ω ω ).Let P ′ = P ∩ M ′ . By elementarity, the restriction of φ to M ′ is a bijection M ′ → P ′ , and so |P ′ | = ℵ .Let A = ω ω ∩ M ′ . If Y ∈ P ′ , then A Y ∈ M ′ , and therefore A Y ⊆ M ′ (because A Y is countable, and M ′ models (enough of) ZFC ). Hence Y ∈ P ′ implies A Y ⊆ A . Let X = A ω . Then X ≈ ω ω (in particular, X is Polish),and X ⊇ A ωY ⊇ Y for all Y ∈ P ′ . In particular, { Y ∈ P : Y ∩ X = ∅} ⊇ P ′ is uncountable, so P is not skinny. (cid:3) Theorem 3.3.
It is consistent relative to a huge cardinal that cov ( ω ωω ) < par ( ω ωω ) . More precisely, given a model of GCH + ( ℵ ω +1 , ℵ ω ) ։ ( ℵ , ℵ ) andany λ > ℵ ω +1 with cf( λ ) > ω , there is a ccc forcing extension in which ℵ ω +1 = cov ( ω ωω ) < par ( ω ωω ) = λ = c . Proof.
Let V be a model of GCH plus ( ℵ ω +1 , ℵ ω ) ։ ( ℵ , ℵ ). Recall that theexistence of such a model is consistent relative to a huge cardinal.Let P be any ccc forcing poset such that (cid:13) P cov( M ) = c = λ . Forexample, we could take P = Fn( λ, λ Cohenreals, or P could be the standard length- λ iteration forcing MA + c = λ .Because P has the ccc, P preserves cardinals; hence (cid:13) P ℵ ω +1 < λ = c .Also, (cid:13) P cf (cid:0) [ ω ω ] ℵ , ⊆ (cid:1) = ℵ ω +1 because P is ccc and the ground modelsatisfies GCH . (This fact is fairly well known, but we include a sketch of theargument here for completeness. By the comments following Corollary 2.3,cf (cid:0) [ ω ω ] ℵ , ⊆ (cid:1) > ℵ ω . Because P has the ccc, every member of [ ω ω ] ℵ in theextension is contained in some member of [ ω ωω ] ℵ ∩ V . That is, [ ω ω ] ℵ ∩ V is cofinal in [ ω ω ] ℵ . Because GCH holds in V , this means cf (cid:0) [ ω ω ] ℵ , ⊆ (cid:1) ≤ (cid:12)(cid:12) [ ω ω ] ℵ ∩ V (cid:12)(cid:12) = ℵ ω +1 .) By Theorem 2.2, (cid:13) P cov ( ω ωω ) = ℵ ω +1 .Furthermore, ( ℵ ω +1 , ℵ ω ) ։ ( ℵ , ℵ ) is preserved by ccc forcing. (Thisfact is considered folklore, but a proof can be found in [7, Lemma 13].) WILL BRIAN
Hence (cid:13) P par ( ω ωω ) ≥ cov( M ) = c by the previous two lemmas. Finally, par ( ω ωω ) ≤ | ω ωω | = | ω ωω | ℵ ≤ c ℵ = c in the extension (because we maypartition ω ωω into singletons). (cid:3) Corollary 3.4.
It is consistent relative to a huge cardinal that c is arbitrarilylarge, and cov ( X ) < par ( X ) for every completely metrizable space X with ℵ ω ≤ wt ( X ) < c such that X contains a subspace homeomorphic to ω ωω .Proof. In statement of Theorem 3.3, let us add the requirement that λ isnot the successor of a singular cardinal with cofinality ω .In the generic extension, let X be a completely metrizable space with wt ( X ) = κ , where ℵ ω ≤ κ < λ = c . By Theorem 2.2, cov ( X ) ≤ cf (cid:0) [ κ ] ℵ , ⊆ (cid:1) .As in the proof of Theorem 3.3 above, [ κ ] ℵ ∩ V is cofinal in cf (cid:0) [ κ ] ℵ , ⊆ (cid:1) ,and therefore cf (cid:0) [ κ ] ℵ , ⊆ (cid:1) ≤ (cid:12)(cid:12) [ κ ] ℵ ∩ V (cid:12)(cid:12) = ( κ ℵ ) V < λ . (The last inequalityuses the fact that GCH holds in V , plus the assumption that λ is not thesuccessor of a singular cardinal with cofinality ω .) Hence cov ( X ) < c . Butif Y is any subspace of X homeomorphic to ω ωω and P is any partition of X into Polish spaces, then { Z ∩ Y : Z ∈ P} is a partition of Y into Polishspaces. Therefore par ( X ) ≥ par ( Y ) = par ( ω ωω ) = c . (cid:3) This raises the question of what completely metrizable spaces with weight ≥ ℵ ω contain a copy of ω ωω . A characterization is given at the beginning ofthe next section: roughly, X contains a (closed) copy of ω ωω unless X andall its closed subsets are everywhere “locally” of weight < ℵ ω .A collection F of countable sets is called sparse if S G is uncountable forevery uncountable G ⊆ F . Equivalently, F is sparse if no single countableset contains uncountably many members of F .Sparse cofinal subsets of (cid:0) [ ω ω ] ℵ , ⊆ (cid:1) have appeared in diverse problemsfrom infinite combinatorics: a good deal about them can be found in Ko-jman, Milovich, and Spadaro’s [17], and they appear also in Spadaro [21],Blass [1], and Nyikos [19]. (Nyikos refers to them as Kuprepa families .) It isknown that ( ℵ ω +1 , ℵ ω ) ։ ( ℵ , ℵ ) implies the non-existence of sparse cofinalfamilies in (cid:0) [ ω ω ] ℵ , ⊆ (cid:1) , and (cid:3) ℵ ω implies they do exist [17, Section 3]. Observation.
If there is a skinny partition of ω ωω into Polish spaces, thenthere is a sparse cofinal subset of (cid:0) [ ω ω ] ℵ , ⊆ (cid:1) . Proof.
Suppose P is a partition of ω ωω into Polish spaces. In the proof ofTheorem 2.2, we showed that this implies D = {{ x ( n ) : x ∈ X and n ∈ ω } : X ∈ P} is a cofinal subset of (cid:0) [ ω ω ] ℵ , ⊆ (cid:1) .Suppose D is not sparse. Then there is some A ∈ [ ω ω ] ℵ such that { B ∈ D : B ⊆ A } is uncountable. By the definition of D , this means that { X ∈ P : X ⊆ A ω } is uncountable, and in particular { X ∈ P : X ∩ A ω = ∅} is uncountable. (cid:3) OVERING VERSUS PARTITIONING WITH POLISH SPACES 9
In other words, a skinny partition of ω ωω is just a special kind of sparsecofinal family, in disguise. In light of this, our use of ( ℵ ω +1 , ℵ ω ) ։ ( ℵ , ℵ )in this section, and our use of weak (cid:3) -like principles in the next, are un-surprising. We note (without proof) that the partitions whose existence isimplied in the proof of Theorem 2.5 or in the next section, which witness“small” values of par ( X ), are skinny, and therefore naturally give rise tosparse cofinal families.4. L -like principles imply cov = par In this section we show that if cov ( X ) < par ( X ) for any completely metriz-able space X , then 0 † exists. We do this by extending the inductive argu-ments of Section 2 to spaces of weight ≥ ℵ ω using “ L -like” assumptions thathold if 0 † does not exist. Roughly, the assumptions we use are that thecardinals cf (cid:0) [ κ ] ℵ , ⊆ (cid:1) are as small as possible, and that a weak (cid:3) principleholds at singular cardinals < c of cofinality ω .First we need a purely topological theorem that makes no assumptionsbeyond ZFC .Let us say that a space X is locally < κ -like if for every nonempty open U ⊆ X , there is a nonempty open V ⊆ U with wt ( V ) < κ . Equivalently, X is locally < κ -like if X has a π -base of open sets with weight < κ .On the other hand, let us say that a space X is locally ≥ κ -like if everynonempty open U ⊆ X has wt ( U ) ≥ κ . Note that X fails to be locally < κ -like if and only if some nonempty open subset of X is locally ≥ κ -like.Compare the following theorem with Theorem 2.4. Theorem 4.1.
Let κ be an uncountable cardinal with cf( κ ) = ω , and let X be a completely metrizable space with wt ( X ) ≥ κ . Then either:(1) some closed subset of X is homeomorphic to κ ω , or(2) there is a partition of X into at most wt ( X ) completely metrizablespaces, each of weight < κ .Furthermore, these two possibilities are mutually exclusive, and (2) holds ifand only if every closed subset of X is locally < κ -like.Proof. To prove the theorem, we show:( a ) If some closed Y ⊆ X is not locally < κ -like, then some closed Z ⊆ Y is homeomorphic to κ ω .( b ) If every closed subspace of X is locally < κ -like, then X can bepartitioned into at most wt ( X ) completely metrizable spaces, eachof weight < κ .We begin with ( a ). Suppose X ⊆ X is closed and is not locally < κ -like.This means there is a nonempty open X ⊆ X (open in X , not necessarilyin X ) that is locally ≥ κ -like. Let Y = X . Note that Y is locally ≥ κ -likebecause X is; in fact, if U is any nonempty open subset of Y , then U islocally ≥ κ -like. Recall that a collection C of subsets of some topological space is discrete if every point has a neighborhood meeting ≤ C . Claim.
Suppose Y is a complete metric space and is locally ≥ κ -like, andlet ε > . There is a discrete collection of κ open subsets of Y , each withdiameter ≤ ε .Proof of claim: By the Bing metrization theorem, Y has a σ -discrete base:that is, there is a sequence hB n : n ∈ ω i of collections of open subsets of Y such that S n ∈ ω B n is a base for Y and each B n is discrete. By deleting anymember of any B n with diameter > ε , we may (and do) assume without lossof generality that each member of each B n has diameter ≤ ε .If some B n has size ≥ κ , we are done; so suppose this is not the case. Fixa discrete collection { U k : k ∈ ω } of open subsets of Y (e.g., any countablesubset of some B n ). For each k and n , let B kn = { U k ∩ V : V ∈ B n } \ {∅} , andnote that S n ∈ ω B kn is a basis for U k . Fix a sequence h µ n : n ∈ ω i of cardinalincreasing up to κ . Because wt ( U k ) ≥ κ , there is some n k ∈ ω such that |B kn k | ≥ µ k . For each k , let C k = (cid:8) V : V ∈ B kn k (cid:9) , and let C = S k ∈ ω C k . (cid:3) By recursion, we now construct a tree { V s : s ∈ κ <ω } of closed subsets of Y as follows. Let V ∅ = Y . Given s ∈ κ <ω , suppose some open set V s hasalready been chosen. Applying the claim above, let { U s ⌢ α : α < κ } be adiscrete collection of open subsets of V s , each of diameter ≤ / | s | . Finally,let V s ⌢ α = U s ⌢ α for each α < κ . This completes the recursion.For each x ∈ κ ω , the completeness of our metric, together with the sizerestriction on the V s , implies that T n ∈ ω V x ↾ n contains exactly one point.Define h : κ ω → Y by taking h ( x ) to be the unique point in T n ∈ ω V x ↾ n foreach x ∈ κ ω .We claim h is a topological embedding. For injectivity, if x = y then x ↾ n = y ↾ n for some n , and x and y are members of the disjoint sets V x ↾ n and V y ↾ n . Continuity follows from our size restriction on the V s : if ε > x ∈ ω ωω , there is some n such that V x ↾ n ⊆ B ε ( h ( x )). Finally, h is aclosed mapping: if [[ s ]] = { x ∈ κ ω : s ⊆ x } is a basic closed set in κ ω , then h ([[ s ]]) = V s ∩ h ( κ ω ). This completes the proof of ( a ).To prove ( b ), suppose every closed K ⊆ X is locally < κ -like. Let W be afunction choosing witnesses to this property: i.e., if K is a nonempty closedsubset of X , then W ( K ) is a nonempty open subset of K with weight < κ .Let λ = wt ( X ). Define a decreasing sequence h X α : α < λ + i of closedsubspaces of X as follows: let X = X , let X α = T ξ<α X ξ for limit α < λ + ,and at successor stages let X α +1 = X α \ W ( X α ).Because wt ( X ) = λ , there is some α < λ + such that X β = X α for all β ≥ α . But this implies X β = ∅ for all β ≥ α , since our construction wouldotherwise give W ( X α ) = ∅ and therefore X α +1 = X α .Each W ( X ξ ) is an open subspace of a closed subspace of X , and there-fore is completely metrizable. Furthermore wt ( W ( X ξ )) < κ for all ξ , and { W ( X ξ ) : ξ < α } is a partition of X \ X α = X . (cid:3) OVERING VERSUS PARTITIONING WITH POLISH SPACES 11
Corollary 4.2.
Let κ be an uncountable cardinal with cf( κ ) = ω , and let X be a completely metrizable space with wt ( X ) = κ . If some closed subset of X is not locally < κ -like, then cov ( X ) = cf (cid:0) [ κ ] ℵ , ⊆ (cid:1) .Proof. Let Z be a closed subspace of X homeomorphic to κ ω . If C is acovering of X with Polish spaces, then { Y ∩ Z : Y ∈ C} is a covering of Z with Polish spaces. Hence cov ( X ) ≥ cov ( Z ) = cov ( κ ω ) = cf (cid:0) [ κ ] ℵ , ⊆ (cid:1) . (Thelast equality is from Theorem 2.2.) But we also have cov ( X ) ≤ cf (cid:0) [ κ ] ℵ , ⊆ (cid:1) by Theorem 2.2, so cov ( X ) = cf (cid:0) [ κ ] ℵ , ⊆ (cid:1) . (cid:3) Corollary 4.3.
Let X be a completely metrizable space with wt ( X ) = ℵ ω .Then either:(1) cov ( X ) = cov ( ω ωω ) and par ( X ) ≥ par ( ω ωω ) .(2) cov ( X ) = par ( X ) = ℵ ω .Proof. We show that statements (1) and (2) from Theorem 4.1 imply, re-spectively, the statements (1) and (2) in the statement of the corollary.Suppose statement (1) from Theorem 4.1 holds, and let Z be a closedsubspace of X homeomorphic to ω ωω . By the previous corollary, cov ( X ) = cov ( ω ωω ) = cf (cid:0) [ ω ω ] ℵ , ⊆ (cid:1) . If P is a partition of X into Polish spaces, then { P ∩ Z : P ∈ P} is a partition of Z into Polish spaces, and this shows that par ( X ) ≥ par ( Z ) = par ( ω ωω ).Suppose statement (2) from Theorem 4.1 holds, and let P be a partitionof X into ≤ ℵ ω completely metrizable spaces of weight < ℵ ω . For each Y ∈ P , there is by Theorem 2.5 a partition P Y of P into < ℵ ω Polishspaces. Then S {P Y : Y ∈ P} is a partition of X into ≤ ℵ ω Polish spaces.Therefore cov ( X ) ≤ par ( X ) ≤ ℵ ω . The reverse inequalities are provided byTheorem 2.2. (cid:3) The next theorem improves Corollary 2.6 from Section 2.
Theorem 4.4.
Suppose cov ( X ) < par ( X ) for some completely metrizablespace X . Then c ≥ ℵ ω +2 .Proof. If wt ( X ) < ℵ ω then cov ( X ) = par ( X ) = wt ( X ) by Theorem 2.5. If wt ( X ) > c then | X | > c (by [6, Theorem 4.1.15]), so cov ( X ) = par ( X ) = | X | .Now suppose wt ( X ) = c . Because c ≤ cf (cid:0) [ c ] ℵ , ⊆ (cid:1) ≤ c ℵ = c , Theo-rem 2.2 gives cov ( X ) = c . Also, c ≤ | X | ≤ c ℵ = c (generally speaking, wt ( Y ) ≤ | Y | ≤ wt ( Y ) ℵ for any metric space Y ). Therefore | X | = c , andthis implies par ( X ) ≤ c (because we may partition X into singletons). Hence c = cov ( X ) ≤ par ( X ) ≤ c , and in particular cov ( X ) = par ( X ).Therefore, if there is a completely metrizable space X with cov ( X ) < par ( X ), then ℵ ω ≤ wt ( X ) < c . To prove the theorem, it suffices to showthat if wt ( X ) = ℵ ω and c = ℵ ω +1 , then cov ( X ) = par ( X ).Assume wt ( X ) = ℵ ω and c = ℵ ω +1 . By Corollary 4.3, we may alsoassume cov ( X ) = cov ( ω ωω ) and par ( X ) ≥ par ( ω ωω ). But | X | ≤ ℵ ℵ ω ≤ c ℵ = c ,hence par ( X ) ≤ c (by partitioning X into singletons). Therefore ℵ ω +1 ≤ cf (cid:0) [ ω ω ] ℵ , ⊆ (cid:1) = cov ( ω ωω ) = cov ( X ) ≤ par ( X ) ≤ c = ℵ ω +1 . (cid:3) The ℵ ω +2 bound given in this theorem is optimal, which can be seen bysetting λ = ℵ ω +2 in Theorem 3.3.Corollary 4.3 gives us our first inkling of how to extend the results of Sec-tion 2 to spaces of weight ℵ ω and beyond. If we assume cf (cid:0) [ ω ω ] ℵ , ⊆ (cid:1) = ℵ ω +1 and we can somehow prove that par ( X ) ≤ ℵ ω +1 whenever wt ( X ) = ℵ ω ,then using Corollary 4.3 we can conclude that cov ( X ) = par ( X ) whenever wt ( X ) = ℵ ω . This is precisely our strategy. Proving par ( X ) ≤ ℵ ω +1 when-ever wt ( X ) = ℵ ω , and similarly for other cardinals of cofinality ω , is where (cid:3) -like principles enter the picture.Let SCH + abbreviate the following statement: SCH + : If κ is a cardinal with uncountable cofinality, then cf (cid:0) [ κ ] ℵ , ⊆ (cid:1) = κ .If κ is a singular cardinal with cf( κ ) = ω , then cf (cid:0) [ κ ] ℵ , ⊆ (cid:1) = κ + .The reason for calling this principle SCH + is that the assertion “ SCH + holdsfor all κ > c ” is equivalent to the Singular Cardinals Hypothesis, abbreviated SCH . This is because if κ > c , then cf (cid:0) [ κ ] ℵ , ⊆ (cid:1) = cov ( κ ω ) = | κ ω | ; so SCH + holds for all κ > c if and only if κ ℵ = κ + for all κ > c with cf( κ ) = ω . (Andthis is equivalent to SCH by a theorem of Silver.)In [10, Section 2], Fuchino and L. Soukup introduce a weak version ofJensen’s principle (cid:3) κ denoted (cid:3) ∗∗∗ ω ,κ . As one might guess from the notation,Fuchino and Soukup discuss a more generalized 2-parameter version of thisprinciple, (cid:3) ∗∗∗ λ,κ . We will only need the special case λ = ω , or rather aconsequence of it described below.Recall that (cid:3) ∗ κ denotes the weak square principle at κ . This is a weakeningof (cid:3) κ , and is equivalent to the existence of a special κ + -Aronszajn tree [14]. Lemma 4.5. (Fuchino and Soukup, [10, Lemma 4])
Assume
SCH + , and let κ be a singular cardinal with cf( κ ) = ω . Then (cid:3) ∗ κ implies (cid:3) ∗∗∗ ω ,κ . It is also worth mentioning that Foreman and Magidor in [8] consideredan axiom called the very weak square principle for κ . The very weak squareprinciple for κ implies (cid:3) ∗∗∗ ω ,κ unconditionally.Instead of applying (cid:3) ∗∗∗ ω ,κ directly, we use a consequence of SCH + + (cid:3) ∗∗∗ ω ,κ proved by Fuchino and Soukup in [10], namely the existence of somethingcalled a ( ω , κ )-Jensen matrix.Let p be a set and κ a cardinal, and let θ be a regular cardinal with ω , κ , | tc( p ) | < θ , where tc( p ) denotes the transitive closure of p . Let H θ denote the set of all sets hereditarily smaller than θ , and recall H θ | = ZFC − .A ( ω , κ ) -Jensen matrix over p is a matrix h M α,n : α < κ + , n < ω i of ele-mentary submodels of H θ such that(1) For each α < κ + and n < ω , p ∈ M α,n , κ +1 ⊆ M α,n , and | M α,n | < κ .(2) For each α < κ + , h M α,n : n < ω i is an increasing sequence.(3) If α < κ + and cf( α ) > ω , then there is some n ∗ < ω such that, forevery n ≥ n ∗ , [ M α,n ] ℵ ∩ M α,n is cofinal in (cid:0) [ M α,n ] ℵ , ⊆ (cid:1) .(4) For each α < κ + , let M α = S n<ω M α,n . Then h M α : α < κ + i iscontinuously increasing, and κ + ⊆ S α<κ + M α . OVERING VERSUS PARTITIONING WITH POLISH SPACES 13
The following is proved as Theorems 7 and 8 in [10]:
Lemma 4.6. (Fuchino and Soukup, [10, Theorems 7 and 8])
Assume
SCH + ,and let κ be a singular cardinal with cf( κ ) = ω . Then (cid:3) ∗∗∗ ω ,κ holds if andonly if there is a ( ω , κ ) -Jensen matrix over p for some (or any) p . We are now ready to prove the main technical result of this section. Com-pare the following with Theorems 2.4 and 4.1.
Theorem 4.7.
Assume
SCH + and (cid:3) ∗∗∗ ω ,κ , and let κ be a singular cardinalwith cf( κ ) = ω . If X is any completely metrizable space of weight κ , thereis a partition of X into ≤ κ + completely metrizable spaces of weight < κ .Proof. Let X be a completely metrizable space of weight κ . ApplyingLemma 2.1, fix a basis B for X such that |B| = κ and every point of X iscontained in only countably many members of B . Applying SCH + , let D bea cofinal family in (cid:0) [ B ] ℵ , ⊆ (cid:1) with |D| = κ + , and fix a bijection p : κ + → D .Let h M α,n : α < κ + , n < ω i be a ( ω , κ )-Jensen matrix over p . As in item(4) above, let M α = S n ∈ ω M α,n for every α < κ + .For each α < κ + and n < ω , let X α,n = { x ∈ X : { U ∈ B : x ∈ U } ⊆ M α,n } and let Y α,n = X α,n \ S { X β,m : either β < α, or else β = α and m < n } . We claim that { Y α,n : α < κ + , n < ω } is a partition of X into ≤ κ + com-pletely metrizable spaces, each of weight < κ .It is clear from the definitions that { Y α,n : α < κ + , n < ω } is a partition of S { X α,n : α < κ + , n < ω } , and that each Y α,n has { U ∩ Y α,n : U ∈ B ∩ M α,n } as a basis, and therefore wt ( Y α,n ) ≤ | M α,n | < κ . To finish the proof of thetheorem, it remains to show that S { X α,n : α < κ + , n < ω } = X and thateach Y α,n is completely metrizable.Let x ∈ X . Then { U ∈ B : x ∈ U } is countable (by our choice of B ), andthere is some A ∈ D with { U ∈ B : x ∈ U } ⊆ A (by our choice of D ). Thereis some γ < κ with p ( γ ) = A , and by (4) there is some α < κ + and n < ω with γ ∈ M α,n . As p ∈ M α,n by (1), we have p ( γ ) = A ∈ M α,n and therefore A ⊆ M α,n . (The “therefore” part uses the elementarity of M α,n in H θ , whichimplies countable members of M α,n are subsets of M α,n .) Hence x ∈ X α,n .As x was arbitrary, it follows that S { X α,n : α < κ + , n < ω } = X .It remains to show each Y α,n is completely metrizable. For each α < κ + ,define X α = S n<ω X α,n . Observe that each X α,n is closed in X , and each X α is therefore an F σ subset of X .Fix α < κ + and n < ω . We consider three cases.First, suppose α = γ + 1 is a successor ordinal. By (4), S { M β,m : β < α, m < ω } = S β ≤ γ M β = M γ = S { M γ,m : m < ω } . From this and our definitions, it follows that S { X β,m : β < α, m < ω } = S { X γ,m : m < ω } = X γ . Consequently, Y α,n = X α,n \ S { X β,m : either β < α, or else β = α and m < n } = X α,n \ (cid:0) X γ ∪ S { X α,m : m < n } (cid:1) . Because X γ is an F σ and each of the X α,m is closed, this shows that Y α,n isa G δ subset of X .Next, suppose α is a limit ordinal with cf( α ) = ω . Fix an increasingsequence h γ k : k < ω i of ordinals with limit α . Arguing in a way similar tothe previous paragraph, (4) implies S { M β,m : β < α, m < ω } = S k ∈ ω M γ k = S { M γ k ,m : k, m < ω } . From this and the definition of the X β,m , it follows that S { X β,m : β < α, m < ω } = S { X γ k ,m : k, m < ω } = S k<ω X γ k . Consequently, just as in the previous case, Y α,n = X α,n \ S { X β,m : either β < α, or else β = α and m < n } = X α,n \ (cid:0) S k<ω X γ k ∪ S { X α,m : m < n } (cid:1) , and this shows that X α,n is a G δ subset of X .Finally, suppose α is a limit ordinal with cf( α ) > ω . By (3), there issome n ∗ < ω such that, for every m ≥ n ∗ , [ M α,m ] ℵ ∩ M α,m is cofinal in (cid:0) [ M α,m ] ℵ , ⊆ (cid:1) . Let k = max { n, n ∗ } . Fix x ∈ X α,n . By (2), M α,n ⊆ M α,k and therefore x ∈ X α,k ; by definition, this means { U ∈ B : x ∈ U } ⊆ M α,n .By our choice of B , { U ∈ B : x ∈ U } is countable. Because k ≥ n ∗ , there is acountable A ∈ M a,n such that { U ∈ B : x ∈ U } ⊆ A . Now A ∈ M α,n implies A ∈ M α , and by (4) (specifically, the “continuously increasing” part), thereis some β < α such that A ∈ M β . In particular, this means A ∈ M β,m forsome m < ω . By the elementarity of M β,m in H θ and the countability of A , A ∈ M β,m implies { U ∈ B : x ∈ U } ⊆ A ⊆ M β,m , which implies x ∈ X β,m .Hence x / ∈ Y α,n . Because x was an arbitrary member of X α,n , this shows Y α,n = ∅ . In particular, Y α,n is Polish. (cid:3) Theorem 4.8.
Suppose
SCH + holds, and (cid:3) ∗∗∗ ω ,µ holds for every singular µ with cf( µ ) = ω . Then for any uncountable cardinal κ , ◦ If cf( κ ) > ω , then cov ( X ) = par ( X ) = κ . ◦ If cf( κ ) = ω , then either ◦ every closed subspace of X is locally < κ -like, in which case cov ( X ) = par ( X ) = κ , or ◦ some closed subspace of X is not locally < κ -like, in which case cov ( X ) = par ( X ) = κ + .In particular, cov ( X ) = par ( X ) for every completely metrizable space X .Proof. Recall that Theorem 2.2 states κ ≤ cov ( X ) ≤ par ( X ) ≤ cf (cid:0) [ κ ] ℵ , ⊆ (cid:1) whenever X is a completely metrizable space of weight κ . Therefore SCH +OVERING VERSUS PARTITIONING WITH POLISH SPACES 15 implies cov ( X ) = par ( X ) = κ whenever wt ( X ) = κ and cf( κ ) > ω . Itremains to prove the cf( κ ) = ω case of the theorem.For this case, note that Corollary 4.2 and SCH + combine to give us thelower bounds we need. Specifically: if κ is an uncountable cardinal withcf( κ ) = ω , then for every completely metrizable space X of weight κ , ◦ κ ≤ cov ( X ) ≤ par ( X ). ◦ If some closed subspace of X is not locally < κ -like, then κ + =cf (cid:0) [ κ ] ℵ , ⊆ (cid:1) = cov ( X ) ≤ par ( X ).To finish the proof, it suffices to show that for every uncountable κ withcf( κ ) = ω , the following statement, which we denote by (IH κ ), holds.(IH κ ) : For every completely metrizable space X of weight κ , ◦ If every closed subspace of X is locally < κ -like, then there is apartition of X into ≤ κ Polish spaces. ◦ If some closed subspace of X is not locally < κ -like, then there is apartition of X into ≤ κ + Polish spaces.For convenience, if cf( κ ) > ω then we let (IH κ ) denote the assertion that cov ( X ) = par ( X ) = κ whenever X has weight κ . As noted above, we knowalready that (IH κ ) is true for all κ with cf( κ ) > ω .We prove (IH κ ) holds for all uncountable κ by transfinite induction on κ .The only yet unproven steps in the induction are limit steps of countablecofinality. So fix κ > ℵ with cf( κ ) = ω , and suppose (IH µ ) holds for alluncountable µ < κ . Let X be a completely metrizable space with wt ( X ) = κ . There are two cases.First, suppose every closed subset of X is locally < κ -like. By Theo-rem 4.1, there is a partition P of X into ≤ κ completely metrizable spaceseach of weight < κ . Because (IH µ ) holds for all uncountable µ < κ , foreach Y ∈ P there is a partition P Y of Y into ≤ κ Polish spaces. Then S {P Y : Y ∈ P} is a partition of X into ≤ κ Polish spaces.Next, suppose some closed subspace of X is not locally < κ -like. ByTheorem 4.7, there is a partition P of X into ≤ κ + completely metrizablespaces, each of weight < κ . Because (IH µ ) holds for all uncountable µ < κ ,for each Y ∈ P there is a partition P Y of Y into ≤ κ Polish spaces. Then S {P Y : Y ∈ P} is a partition of X into ≤ κ + Polish spaces. (cid:3)
As noted in the introduction, this theorem is only really interesting inthe case κ ≤ c . However, this restriction is not required at any point in theproof and is omitted intentionally from the statement of Theorem 4.8. Itis worth pointing out that, with a little more work, one need only assume (cid:3) ∗∗∗ µ when cf( µ ) = ω and ℵ ω ≤ µ < c . The reason is that if κ > c then cov ( X ) = par ( X ) = | X | : then SCH + , Theorem 4.1, and [6, Theorem 4.1.15]allow us to compute that | X | = κ , unless cf( κ ) = ω and some closed subspaceof X is not locally < κ -like, in which case | X | = κ ℵ = κ + . Theorem 4.9.
Suppose there is some completely metrizable space X with cov ( X ) < par ( X ) . Then † exists.Proof. If (cid:3) ∗∗∗ κ fails for a singular cardinal κ , then (cid:3) κ fails also by Lemma 4.5.The failure of (cid:3) κ for a singular cardinal κ implies the existence of 0 † , andin fact much more, by results of Cummings and Friedman [3].Therefore, by the previous theorem, it suffices to show that the failure of SCH + fails implies 0 † exists.By an inner model we mean a (definable, with parameters) class M suchthat M | = ZFC . An inner model of
GCH means additionally M | = GCH .We say the Covering Lemma holds over an inner model M if for everyuncountable X ⊆ M , there is some Y ∈ M with Y ⊇ X such that | X | = | Y | .It is proved in [16] (cf. Lemmas 4.9 and 4.10) that if the Covering Lemmaholds over an inner model of GCH , then
SCH + holds.By work of Dodd and Jensen [4], if there is no inner model containing ameasurable cardinal, then the Covering Lemma holds over an inner modelof GCH (specifically, the Dodd-Jensen core model K DJ ). By further workof Dodd and Jensen [5], if there is an inner model containing a measurablecardinal but 0 † does not exist, then the Covering Lemma holds over an innermodel of GCH (specifically, either L [ U ] for some measure U with crit ( U ) assmall as possible, or L [ U, C ] for some C Prikry-generic over L [ U ]).Either way, if 0 † does not exist then Jensen’s Covering Lemma holds overan inner model M of GCH , and
SCH + follows by [16]. (cid:3) This argument leaves open the precise consistency strength of ¬ SCH + . Question 4.10. Is ¬ SCH + equiconsistent with the existence of a measurablecardinal µ of Mitchell order µ ++ ? By work of Gitik [11, 12], the failure of
SCH is equiconsistent with the ex-istence of a measurable cardinal µ of Mitchell order µ ++ . An affirmativeanswer to this question would raise the consistency strength of the strict in-equality cov ( X ) < par ( X ) to measurable cardinals of high Mitchell order (cf.[3]). On the other side of things, it would also be interesting to reduce theupper bound from Section 3 on the consistency strength of this statement. Question 4.11.
Beginning with a supercompact cardinal, can one force cov ( X ) < par ( X ) for some completely metrizable space X ? Acknowledgments
I would like to thank Alan Dow for helpful discussions about some ofthe ideas in this paper, and Alexander Osipov for raising some questionsrelated to the material in Section 2. I would also like to thank Yair Hayutfor his observations in [13]. An early version of the proof of Theorem 4.7used a much stronger L -like hypothesis, and Hayut proved the negation ofthis hypothesis equiconsistent with the existence of a single inaccessible.This inspired a search for a more efficient proof, eventually raising the lowerbound on the consistency strength of cov ( X ) < par ( X ) to 0 † . OVERING VERSUS PARTITIONING WITH POLISH SPACES 17
References [1] A. Blass, “On the divisible parts of quotient groups,” G¨obel, R¨udiger (ed.) et al.,
Abelian group theory and related topics . Conference, August 1993, Oberwolfach. Prov-idence, RI: American Mathematical Society. Contemp. Math. (1994), pp. 37–50.[2] W. Brian and A. W. Miller, “Partitions of 2 ω and completely ultrametrizable spaces,” Topology and its Applications (2015), pp. 61–71.[3] J. Cummings and S. D. Friedman, “ (cid:3) on the singular cardinals,”
Journal of SymbolicLogic (2008), pp. 1307–1314.[4] A. Dodd and R. Jensen, “The covering lemma for K,” Annals of Mathematical Logic , vol. 1 (1982), pp. 1–30.[5] A. Dodd and R. Jensen, “The covering lemma for L [ U ],” Annals of MathematicalLogic , vol. 2 (1982), pp. 127–135.[6] R. Engelking, General Topology.
Sigma Series in Pure Mathematics, 6, Heldermann,Berlin (revised edition), 1989.[7] M. Eskew and Y. Hayut, “On the consistency of local and global versions of Chang’sConjecture,”
Transactions of the American Mathematical Society (2018), pp.2879–2905.[8] M. Foreman and M. Magidor, “A very weak square principle,”
Journal of SymbolicLogic (1997), pp. 175–196.[9] D. H. Fremlin and S. Shelah, “On partitions of the real line,” Israel Journal ofMathematics (1979), pp. 299–304.[10] S. Fuchino and L. Soukup, “More set-theory around the weak Freese-Nation prop-erty,” Fundamenta Mathematicae (1997), pp. 159–176.[11] M. Gitik, “The negation of
SCH from o ( κ ) = κ ++ ,” Annals of Pure and Applied Logic (1989), pp. 209–234.[12] M. Gitik, “The strength of the failure of the Singular Cardinal Hypothesis,” Annalsof Pure and Applied Logic (1991), pp. 215–240.[13] Y. Hayut, “Getting a model of ZFC that fails to nicely cover an inner model,” Math-Overflow answer (2020), mathoverflow.net/questions/377236 .[14] R. Jensen, “The fine structure of the constructible hierarchy,”
Annals of MathematicalLogic (1972), pp. 229–308.[15] A. Kechris, Classical Descriptive Set Theory , Graduate Texts in Mathematics, vol.156, Springer-Verlag, 1995.[16] W. Just, A. R. D. Mathias, K. Prikry, and P. Simon, “On the existence of large p -ideals,” Journal of Symbolic Logic (1990), pp. 457–465.[17] M. Kojman, D. Milovich, and S. Spadaro, “Noetherian type in topological products,” Israel Journal of Mathematics (2014), pp. 195–225.[18] J. P. Levinski, M. Magidor, and S. Shelah, “Chang’s Conjecture for ℵ ω ,” Israel Jour-nal of Mathematics (1990), pp. 161–172.[19] P. Nyikos, “Generalized Kurepa and MAD families and topology,” unpublished noteavailable at https://people.math.sc.edu/nyikos/Ku.pdf .[20] A. Sharon and M. Vialle, “Some consequences of reflection on the approachabilityideal,” Transactions of the American Mathematical Society (2009), pp. 4201–4214.[21] S. Spadaro, “On two topological cardinal invariants of an order-theoretic flavour,”
Annals of Pure and Applied Logic , No. 12 (2012), pp. 1865–1871.
W. R. Brian, Department of Mathematics and Statistics, University ofNorth Carolina at Charlotte, 9201 University City Blvd., Charlotte, NC28223, USA
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