aa r X i v : . [ m a t h . M G ] A ug APD profiles and transfinite asymptotic dimension
Kamil OrzechowskiFaculty of Mathematics and Natural SciencesUniversity of Rzesz´ow35-959 Rzesz´ow, PolandE-mail: [email protected] 2, 2019
Abstract
We develop the theory of APD profiles introduced by J. Dydak for ∞ -pseudometric spaces([3]). We connect them with transfinite asymptotic dimension defined by T. Radul ([4]). We givea characterization of spaces with transfinite asymptotic dimension at most ω + n for n ∈ ω anda sufficient condition for a space to have transfinite asymptotic dimension at most m · ω + n for m, n ∈ ω , using the language of APD profiles. Keywords : asymptotic dimension, asymptotic property C, APD profile. : 54F45.
1. The set-theoretical background
We begin with a set-theoretical definition due to P. Borst ([1, Chapter II]). Let L be an arbitraryset and Fin L the family of all finite, non-empty subsets of L . For any M ⊂ Fin L and σ ∈ {∅} ∪ Fin L we put M σ = { τ ∈ Fin L : τ ∪ σ ∈ M and τ ∩ σ = ∅} . We abbreviate M { a } by M a for any a ∈ L . Definition 1.1.
Let M be a subfamily of Fin L . Define the ordinal number Ord M inductively asfollows:Ord M = 0 iff M = ∅ ;Ord M ≤ α iff Ord M a < α for every a ∈ L ;Ord M = α iff Ord M ≤ α and Ord M < α is not true;Ord M = ∞ iff Ord M > α for every ordinal number α .In the case Ord M = ∞ we can also say that the ordinal number Ord M does not exist . Werecollect some basic properties of the ordinal number Ord M . Lemma 1.2 ([1, 2.1.4]) . Let L be a set and M ⊂ Fin L . In addition, let n ∈ ω . Then Ord M ≤ n ifand only if | σ | ≤ n for every σ ∈ M . Thus one can say that Ord M is a transfinite generalization of the supremum of cardinalities ofall members of M .We call a subfamily M of Fin L inclusive iff for every σ, σ ′ ∈ Fin L such that σ ′ ⊂ σ : σ ∈ M implies σ ′ ∈ M . By N we denote the set of all positive integers.1 emma 1.3 ([1, 2.1.3]) . Let L be a set and M an inclusive subfamily of Fin L . Then Ord M = ∞ iffthere exists a sequence ( a i ) i ∈ N of distinct elements of L such that σ n = { a i : 1 ≤ i ≤ n } ∈ M for each n ∈ N . Lemma 1.4 ([1, 2.1.6]) . Let φ : L → L ′ be a function and M ⊂ Fin L , M ′ ⊂ Fin L ′ be such that forevery σ ∈ M we have φ ( σ ) ∈ M ′ and | φ ( σ ) | = | σ | . Then Ord M ≤ Ord M ′ . Lemma 1.5 (cf. [4, Theorem 4]) . If L is a countable set, M ⊂ Fin L and Ord
M < ∞ , then Ord
M < ω (which means it is a countable ordinal number). We will prove another useful lemma (one direction comes from [1, 2.1.5]).
Lemma 1.6.
Let L be a set, M ⊂ Fin L and γ ∈ {∅} ∪ Fin L . Let α > be an ordinal number and p ∈ ω . Then Ord M γ < α + p if and only if Ord M γ ∪ σ < α for every σ ∈ {∅} ∪ Fin L with | σ | = p and γ ∩ σ = ∅ .Proof. We proceed by induction on p . If p = 0, then σ must be empty and the assertion is trivial.Assume the lemma holds for p −
1. Since α + p = ( α + 1) + ( p − M γ < α + p if andonly if Ord M γ ∪ σ ′ < α + 1 for every σ ′ with | σ ′ | = p − σ ′ ∩ γ = ∅ .Suppose the latter and take an arbitrary σ ∈ {∅} ∪ Fin L with | σ | = p and γ ∩ σ = ∅ . Pick anelement a ∈ σ and let σ ′ := σ \ { a } . We have Ord M γ ∪ σ = Ord ( M γ ∪ σ ′ ) a , which is by definition lessthan α .Suppose now that Ord M γ ∪ σ < α for every σ ∈ {∅} ∪ Fin L with | σ | = p and γ ∩ σ = ∅ . Take anarbitrary σ ′ with | σ ′ | = p − σ ′ ∩ γ = ∅ . We want to show that Ord M γ ∪ σ ′ ≤ α , i.e. Ord ( M γ ∪ σ ′ ) a < α for any a ∈ L . It suffices to check it only for a γ ∪ σ ′ . Then ( M γ ∪ σ ′ ) a = M γ ∪ σ for σ := σ ′ ∪ { a } and we can use our assumption.The most interesting case is that for γ = ∅ . Corollary 1.7.
Let L be a set, M ⊂ Fin L , α > and p ∈ ω . Then Ord
M < α + p if and only if Ord M σ < α for every σ ⊂ L with | σ | = p . We are going to establish a condition classifying inclusive families M ⊂ Fin L with Ord M < ω · ω .It will be useful to introduce some (simplified) game-theoretical terminology. Definition 1.8.
Let m, n ∈ ω , n >
0. We call a subset S of the cartesian power (Fin L ) m +1 an m - strategy starting at n if and only if it satisfies the following conditions:1. | σ | = n for any ( σ , . . . , σ m ) ∈ S ;2. if ( σ , . . . , σ m ) ∈ S , ( τ , . . . , τ m ) ∈ S and σ i = τ i for 0 ≤ i ≤ k < m , then | σ k +1 | = | τ k +1 | ;3. if 0 ≤ k ≤ m and ( σ , . . . , σ k ) is an initial segment of some ( σ , . . . , σ m ) ∈ S , then so is( σ , . . . , σ k − , τ ) for any τ with | τ | = | σ k | .Loosely speaking, given ( σ , . . . , σ k ) already constructed following S , the strategy determines thecardinality of the next term σ k +1 . Proposition 1.9.
Let L be a set, M an inclusive subfamily of Fin L and m, n ∈ ω . Then Ord M ≤ m · ω + n if and only if there exists an m -strategy S starting at n + 1 such that for every sequence ( σ , σ , . . . , σ m ) ∈ S of pairwise disjoint sets the condition σ ∪ · · · ∪ σ m M holds. roof. We proceed by induction on m ∈ ω . For m = 0 and fixed n ∈ ω , we have Ord M ≤ n iff thecardinality of all σ ∈ M is bounded by n . Since M is inclusive, it is equivalent to say that σ M forany σ with | σ | = n + 1, i.e. the (unique) 0-strategy starting at n + 1 fullfills the requirements of theproposition.Let m > m −
1. Fix n ∈ ω . Assume Ord M ≤ m · ω + n . ByCorollary 1.7 we have Ord M σ < m · ω for any σ with | σ | = n + 1. Thus, Ord M σ ≤ ( m − · ω + n for some n = n ( σ ) ∈ ω . By the inductive assumption, there is an ( m − S ( σ ) starting at n + 1 such that σ ∪ . . . σ m M σ for every sequence ( σ , . . . , σ m ) ∈ S ( σ ) of pairwise disjoint sets.Define S to consist of all ( σ , σ , . . . , σ m ) such that | σ | = n + 1 and ( σ , . . . , σ m ) ∈ S ( σ ). It is nothard to check that S is an m -strategy starting at n + 1 and satisfies σ ∪ · · · ∪ σ m M for pairwisedisjoint ( σ , σ , . . . , σ m ) ∈ S .Suppose we have an m -strategy starting at n + 1 and satisfying σ ∪ · · · ∪ σ m M for pairwisedisjoint ( σ , σ , . . . , σ m ) ∈ S . Then any σ with | σ | = n +1 determines a number n ( σ ) and an ( m − S ( σ ) starting at n ( σ ) + 1 consisting of those ( σ , . . . , σ m ) for which ( σ , σ , . . . , σ m ) ∈ S .Therefore σ ∪ . . . σ m M σ for every sequence ( σ , . . . , σ m ) ∈ S ( σ ) of pairwise disjoint sets. Hence,by the inductive assumption, Ord M σ ≤ ( m − · ω + n ( σ ) < m · ω . Since it holds for any | σ | = n +1,Corollary 1.7 implies Ord M ≤ m · ω + n .
2. Two approaches to transfinite asymptotic dimension
The following definitions are usually formulated for metric spaces. However, we adjust themto the broader context of ∞ - pseudometric spaces , i.e. spaces consisting of a set X with a function d : X × X → [0 , ∞ ] satisfying the properties of symmetry, triangle inequality and taking value 0 onthe diagonal in X × X . Definition 2.1.
We say that a family U of subspaces of an ∞ -pseudometric space ( X, d ) is uniformlybounded if the number mesh( U ) := sup { diam( U ) : U ∈ U } is finite. Let r >
0, we say that U is r - disjoint if for any different A, B ∈ U we havedist(
A, B ) := inf { d ( a, b ) : a ∈ A, b ∈ B } ≥ r. Definition 2.2.
Let (
X, d ) be an ∞ -pseudometric space, x ∈ X and r >
0. The scale- r -component of x in X is the set of all points y ∈ X that can be connected to x by a scale- r -chain, i.e. a sequenceof points y = x , . . . , x n = x in X such that B ( x i , r ) ∩ B ( x i +1 , r ) = ∅ for each 0 ≤ i < n .We say that X is of scale- r -dimension r -components taken for all points x ∈ X is uniformly bounded. More generally, X is of scale- r -dimension at most n if it can berepresented as the union of some n + 1 subspaces of scale- r -dimension 0.We associate with an ∞ -pseudometric space ( X, d ) two inclusive families of finite subsets of N .The first of them is taken from [4]. Definition 2.3.
We define A = A ( X, d ) to consist precisely of all σ ∈ Fin N such that there is nofamily ( X i ) i ∈ σ of subspaces of X which covers X and each X i decomposes as the union of some i -disjoint uniformly bounded family.Similarly, we can define another family. Definition 2.4.
We define M = M ( X, d ) to consist precisely of all σ ∈ Fin N such that there is nofamily ( X i ) i ∈ σ of subspaces of X which covers X and each X i has scale- i -dimension 0.3e have the following Proposition 2.5.
For any ∞ -pseudometric space ( X, d ) : Ord A ( X, d ) = Ord M ( X, d ) .Proof. Observe that if Y is the union of a 2 r -disjoint uniformly bounded family U , then each scale- r -chain in Y must lie in some common U ∈ U , thus every scale- r -component of Y has diameterbounded by mesh( U ). So, if σ ∈ M , then { n : n ∈ σ } ∈ A . Applying Lemma 1.4 to the function φ : N → N , φ ( n ) = 2 n , we conclude that Ord M ≤ Ord A . For the converse, notice that differentscale- r -components are r -disjoint so σ ∈ A implies σ ∈ M and applying Lemma 1.4 to the identityfunction on N we obtain Ord A ≤
Ord M . Definition 2.6 ([4]) . We call the ordinal number Ord A ( X, d ) = Ord M ( X, d ) transfinite asymptoticdimension of X and denote it by trasdim ( X, d ). Definition 2.7.
We say that X has asymptotic property C if and only if trasdim X < ∞ .Using Lemma 1.3 and treating trasdim X < ∞ as Ord A , we get original Dranishnikov’s definition([2]), namely: X has asymptotic property C if and only if for any sequence ( a i ) i ∈ N of distinct naturalnumbers there exists n and a sequence ( U i ) ni =1 of uniformly bounded families such that S ni =1 U i covers X and U i is a i -disjoint for i = 1 , . . . , n .Thinking of trasdim X < ∞ rather as of Ord M , we get definition due to J. Dydak ([3, 5.12]): X has asymptotic property C if and only if for any sequence ( a i ) i ∈ N of distinct natural numbers thereexists n and a decomposition X = S ni =1 X i such that each X i is of scale- a i -dimension 0.
3. APD profiles and transfinite asymptotic dimension
In his paper [3] J. Dydak defined so called
APD profile of an ∞ -pseudometric space. It is justifiedand convenient to deal only with integral APD profiles. Definition 3.1.
Suppose X is an ∞ -pseudometric space. A finite array of non-decreasing functions( α , . . . , α k ) from N to N is an integral APD profile of X if and only if α is constant and for anynon-decreasing array ( r , . . . , r k ) of positive integers there is a decomposition of X as the union of itssubsets X , . . . , X k such that each X i has scale- r i -dimension at most α i ( r i − ) − ≤ i ≤ k and 0for i = 0.It was proved in [3] that having an APD profile is a hereditary coarse invariant and so is theminimal length of APD profiles. A space X has asymptotic dimension at most n iff (1 , n ) is an APDprofile of X . Moreover, X has finite asymptotic dimension iff it has an APD profile consisting ofconstant functions. Spaces which admit an APD profile are said to have asymptotic property D , whichimplies having asymptotic property C. We are interested in finding a deeper relation between the formof an APD profile and the precise value of trasdim X . Theorem 3.2.
Let n ∈ ω . An ∞ -pseudometric space X has an integral APD profile ( n + 1 , f ) if andonly if trasdim X ≤ ω + n .Proof. Suppose that ( n + 1 , f ) is an integral APD of X . By definition, for any r ≤ r there existsa decomposition X = Y ∪ Y such that Y further decomposes as Y = X ∪ · · · ∪ X n +1 and Y as Y = X n +2 ∪ · · · ∪ X n + f ( r )+1 , where each X i is of scale- r -dimension 0 for 1 ≤ i ≤ n + 1 and ofscale- r -dimension 0 for n + 2 ≤ i ≤ n + f ( r ) + 1. Take a subset σ ⊂ N of cardinality n + 1. Listits elements in increasing order: σ = { a , . . . , a n +1 } . We will show that Ord M σ < f ( a n +1 ) < ω .4et τ = { b , . . . , b f ( a n +1 ) } (elements listed in increasing order) be a subset of N disjoint with σ . Weclaim that σ ∪ τ M . Let us take r := a n +1 and r := max (cid:0) a n +1 , b f ( a n +1 ) (cid:1) . Then the family( X i ) n + f ( a n +1 )+1 i =1 witnesses that σ ∪ τ M . Hence all members of M σ have cardinality less than f ( a n +1 ). Using Lemma 1.2 and Corollary 1.7, we finish one part of the proof.Suppose trasdim X ≤ ω + n . For k ∈ N put f ( k ) := Ord M { k,...,k + n } + 1. From Corollary1.7 we conclude that f takes values in N . It is not hard to check that f is non-decreasing. Weclaim that ( n + 1 , f ) is an integral APD profile of X . Fix natural numbers r ≤ r . Consider theset τ = { r , . . . , r + n, m, m + 1 , . . . , m + f ( r ) − } , where m := max ( r , r + n + 1), and let τ ′ = { m, m + 1 , . . . , m + f ( r ) − } . The cardinality of τ ′ exceeds Ord M { r ,...,r + n } so τ ′ M { r ,...,r + n } and τ M . The latter means that there exists a decomposition X = X r ∪· · ·∪ X r + n ∪ X m ∪· · ·∪ X m + f ( r ) − such that each X i has scale- i -dimension 0 for i ∈ τ . In particular, all subspaces X r , . . . , X r + n havescale- r -dimension 0, hence they form one subspace of scale- r -dimension at most n . Similarly, thesubspaces X m , . . . , X m + f ( r ) − form one subspace of scale- r -dimension at most f ( r ) −
1. Thus( n + 1 , f ) is an APD profile of X . Corollary 3.3. An ∞ -pseudometric space satisfies trasdim X < ω + ω if and only if it has an integralAPD profile consisting of two functions ( α , α ) . M. Satkiewicz formulated in [5] the omega conjecture asserting that if ω ≤ trasdim X < ∞ , thentrasdim X = ω . Recently, Y. Wu and J. Zhu ([6]) have disproved it constructing a metric space withtransfinite asymptotic dimension ω + 1. Our theorem implies that this space has an APD profile ofthe form (2 , f ) for some f but does not have an APD profile of the form (1 , g ) for any g .One direction of Theorem 3.2 can be generalized as follows: Theorem 3.4.
Let n, m ∈ ω , m > . If an ∞ -pseudometric space X has an integral APD profile ( n + 1 , α , . . . , α m ) for some functions α , . . . , α m , then trasdim X ≤ m · ω + n .Proof. Define S ⊂ (Fin N ) m +1 to consist of all ( σ , . . . , σ m ) ∈ (Fin N ) m +1 such that | σ | = n + 1 and | σ k | = α k (cid:16) max S k − i =0 σ i (cid:17) for k = 1 , . . . , m . It is easy to check that S is an m -strategy starting at n + 1.According to Proposition 1.9, it is sufficient to show that for every sequence ( σ , σ , . . . , σ m ) ∈ S ofpairwise disjoint sets the condition σ ∪ · · · ∪ σ m M ( X, d ) holds. Fix ( σ , σ , . . . , σ m ) ∈ S withpairwise disjoint terms. Put r k := max S ki =0 σ i for k = 0 , . . . , m . Obviously r ≤ · · · ≤ r m . Applyingthe definition of an APD profile, we obtain a decomposition X = X ∪ · · · ∪ X m such that X k is ofscale- r k -dimension at most α k ( r k − ) − α k (cid:16) max S k − i =0 σ i (cid:17) − | σ k | −
1, for 1 ≤ k ≤ m , andat most n for k = 0. Thus, each X k decomposes further as the union of | σ k | subspaces, each ofthem being of scale- r k -dimension 0. We can write this decomposition as X k = S j ∈ σ k X k,j . Since r k = max S ki =0 σ i , each X k,j is in particular of scale- j -dimension 0. Combining all the X k , we geta decomposition X = S i ∈ σ ∪···∪ σ m X i where every X i is of scale- i -dimension 0. That means that σ ∪ · · · ∪ σ m M ( X, d ), as desired.We have the following
Corollary 3.5.
If an ∞ -pseudometric space X has asymptotic property D, then trasdim X < ω · ω . The strategy S constructed while proving Theorem 3.4 is rather special because the cardinalityof σ k is determined somewhat “uniformly” by given α k (taking an expression of previous σ , . . . , σ k − as its argument). In general, a strategy in Definition 1.8 does not provide such a uniform rule, it ismerely a whole strategy tree of the game in which we react with a natural number to a given sequenceof subsets (and such a reaction could depend very “wildly” on a current position).5 uestion 3.6. Is there a space X with trasdim X < ω · ω but without asymptotic property D? If the answer were affirmative, there would be a space with asymptotic property C but withoutaymptotic property D (thus responding the question in [3, 5.15]).
References [1] P. Borst,
Classification of weakly infinite-dimensional spaces , Fund. Math. 130 (1988), 1–306.[2] A. Dranishnikov,
Asymptotic topology , Russ. Math. Surv. 55 (2000), 1085–1129.[3] J. Dydak,
Matrix algebra of sets and variants of decomposition complexity , arXiv:1612.067771v4.[4] T. Radul,
On transfinite extension of asymptotic dimension , Topology and its Applications 157(2010), 2292–2296.[5] M. Satkiewicz,
Transfinite Asymptotic Dimension , arXiv:1310.1258v1.[6] Y. Wu, J. Zhu,
A metric space with its transfinite asymptotic dimension ωω