The normality of macrocubes and hyperballeans
aa r X i v : . [ m a t h . GN ] J u l THE NORMALITY OF MACROCUBES AND HYPERBALLEANS
IGOR PROTASOV, KSENIA PROTASOVA
Abstract.
For a bornology B on a cardinal κ , we prove that the B -macrocube isnormal if and only if B has a linearly ordered base. As a corollary, we get that thehyperballean of bounded subsets of an ultradiscrete ballean is not normal. Theseanswer Question 1 from [2] and Question 14.4 from [1]. MSC: 54E05, 54E15, 05D10Keywords: coarse structure, ballean, macrocube, hyperballeans, Ramsey ultrafilter.1.
Introduction and preliminaries
Given a set X , a family E of subsets of X × X is called a coarse structure on X if • each E ∈ E contains the diagonal △ X := { ( x, x ) : x ∈ X } of X ; • if E , E ′ ∈ E then E ◦ E ′ ∈ E and E − ∈ E , where E ◦ E ′ = { ( x, y ) : ∃ z (( x, z ) ∈ E, ( z, y ) ∈ E ′ ) } , E − = { ( y, x ) : ( x, y ) ∈ E } ; • if E ∈ E and △ X ⊆ E ′ ⊆ E then E ′ ∈ E .Elements E ∈ E of the coarse structure are called entourages on X For x ∈ X and E ∈ E the set E [ x ] := { y ∈ X : ( x, y ) ∈ E} is called the ball of radius E centered at x . Since E = S x ∈ X { x } × E [ x ], the entourage E is uniquely determinedby the family of balls { E [ x ] : x ∈ X } . A subfamily E ′ ⊆ E is called a base of the coarsestructure E if each set E ∈ E is contained in some E ′ ∈ E .The pair ( X, E ) is called a coarse space [10] or a ballean [7], [9].In this paper, all balleans under consideration are supposed to be connected : for any x, y ∈ X , there is E ∈ E such y ∈ E [ x ]. A subset Y ⊆ X is called bounded if Y = E [ x ]for some E ∈ E , and x ∈ X . The family B X of all bounded subsets of X is a bornologyon X . We recall that a family B of subsets of a set X is a bornology if B contains thefamily [ X ] <ω of all finite subsets of X and B is closed under finite unions and takingsubsets. A bornology B on a set X is called unbounded if X / ∈ B . A subfamily B ′ of B is called a base for B if, for each B ∈ B , there exists B ′ ∈ B ′ such that B ⊆ B ′ .Each subset Y ⊆ X defines a subbalean ( Y, E| Y ) of ( X, E ), where E| Y = { E ∩ ( Y × Y ) : E ∈ E} . A subbalean ( Y, E| Y ) is called large if there exists E ∈ E such that X = E [ Y ],where E [ Y ] = S y ∈ Y E [ y ].Let ( X, E ), ( X ′ , E ′ ) be balleans. A mapping f : X → X ′ is called macrouniform if forevery E ∈ E there exists E ′ ∈ E such that f ( E ( x )) ⊆ E ′ ( f ( x )) for each x ∈ X . If f isa bijection such that f and f − are macrouniform, then f is called an asymorphism . If( X, E ) and ( X ′ , E ′ ) contains large asymorphic subballeans, then they are called coarselyequivalent. Every metric d on a set X defines the coarse structure E d on X with the base {{ ( x, y ) : d ( x, y ) < n } : n ∈ N } . A ballean ( X, E ) is called metrizable if there is a metric d on suchthat E = E d . Theorem 1.1. ([9, Theorem 2.1.1])
A ballean ( X, E ) is metrizable if and only if E has a countable base. Let ( X, E ) be a ballean. A subset U ⊆ X is called an asymptotic neighbourhood of asubset Y ⊆ X if for every E ∈ E the set E [ Y ] \ U is bounded.Two subset Y, Z of X are called asymptotically disjoint (separated) if for every E ∈ E the intersection E [ Y ] ∩ E [ Z ] is bounded ( Y and Z have disjoint asymptotic neighbour-hoods). We say that Y, Z are linked if Y, Z are not asymptotically disjoint.A ballean ( X, E ) is called normal [6] if any two asymptotically disjoint subsets of X are asymptotically separated. Every ballean ( X, E ) with linearly ordered base of E isnormal. In particular, every metrizable ballean is normal, see [6, Proposition 1.1].A function f : X → R is called slowly oscillating if for any E ∈ E and ε >
0, thereexists a bounded subset B of X such that diam f ( E [ x ]) < ε for each x ∈ X \ B . ✷ Theorem 1.2. ([6, Theorem 2.1.])
A ballean ( X, E ) is normal if and only if for anytwo disjoint asymptotically disjoint subsets Y, Z of X there exists a slowly oscillatingfunction f : X → [0 , such that f | Y ≡ and f | Z ≡ . Theorem 1.3. ([1, Theorem 1.4 .])
If the product X × Y of two unbounded balleans X, Y is normal then the bornology B X × Y has a linearly ordered base. Macrocubes and hyperballeans
Let B be a bornology on an infinite cardinal κ such that κ / ∈ B , { , } B = { ( x α ) α<κ ) ∈{ , } κ : { α : x α = 1 } ∈ B . We take the family {{ (( x α ) α<κ , ( y α ) α<κ ) ∈ { , } B × { , } B : x α = y α , α ∈ κ \ B } : B ∈ B} as a base of the coarse structure on { , } B . The obtained ballean is called the B -macrocube [2]. If κ = ω and B = [ ω ] <ω then we get the well-known Cantor macrocubewhose coarse characterization was given in [3].Given a ballean ( X, E ) the hyperballean ( X, E ) ♭ is a ballean on B ( X, E ) endowed witha coarse structure with the base { E ♭ : E ∈ E} , E ♭ [ Y ] = { Z : Y ⊆ E [ Z ], Z ⊆ E [ Y ] } ,see [4], [8].Every bornology B on a cardinal κ defines the discrete ballean ( κ, E B ), E B has thebase { E B : B ∈ B} , E B [ x ] = B if x ∈ B and E B [ x ] = { x } if x ∈ κ \ B . If the set { κ \ B : B ∈ B} is an ultrafilter, then ( κ, E B ) is called ultradiscrete . We denote by [ κ ] B the hyperballean ( κ, E B ) ♭ . Proposition 2.1.
Let κ be a cardinal, B ) be a bornology on κ , K = { A ∈ [ κ ] B :0 ∈ A . Then the characteristic function f : [ κ ] B −→ { , } B is macro-uniform and therestriction of f to K is an asymorphism between K and f ( K ) . HE NORMALITY OF MACROCUBES AND HYPERBALLEANS 3
Proposition 2.2.
For any cardinal κ and bornology B on κ , the balleans { , } B and [ κ ] B are not coarsely equivalent.Proof. Following [3], we say that a ballean ( X, E has asymptotically isolated balls if,for any bounded subset B and E ∈ E there exists x ∈ X \ B such that E ′ [ x ] \ E [ x ] = ∅ .If ( X ′ , E ′ ) is coarsely equivalent to ( X, E ) then ( X ′ , E ′ ) has asymptotically isolated balls.We observe { , } B does not have asymptotically isolated balls, but the subballean[ κ ] B = { A ⊂ κ : | A | = 1 } of [ κ ] B has asymptotically isolated balls. Since [ κ ] B and[ κ ] B \ [ κ ] B are asymptotically disjoint, we see that [ κ ] B has asymptotically isolated balls. ✷ Proposition 2.3.
Let B be a bornology on a cardinal κ such that { κ \ B : B ∈ B} isnot an ultrafilter. Then [ κ ] B is normal if and only if B has a linearly ordered base.Proof. If B has a linearly ordered base then [ κ ] B is normal by [6, Proposition 1.1].We assume that [ κ ] B is normal, partition κ in two unbounded subsets K , K and put B = B| K , B = B| K . Then { , } B is asymorphic to { , } B × { , } B . By Theorem1.3, B has a linearly ordered base. ✷ Some algebra on ultrafilters
Let G be an Abelian group. We endow G with the discrete topology and identify theStone- ˇ C ech compactification βG of G with the family of all ultrafilters on G .For A ⊆ G , we denote ¯ A = {U ∈ βG : A ∈ U } .Then the family { ¯ A : A ⊆ G } forms a base for the topology on βG . Every mapping f : G −→ [0 ,
1] can be extended to the continuous mapping f β : βG −→ [0 , G onto βG by the following rule: for U , V ∈ βG we take U ∈ U and, for each x ∈ U , pick V x ∈ U . Then S x ∈ U ( x + V x ) ∈ U + V ,and the family of all these subsets forms a base of the ultrafilter U + V .4. Results
Theorem 4.1.
Let B be a bornology on a cardinal κ . The B -macrocube X is normalif and only if B has a linearly ordered base.Poof. In light of Proposition 2.3, it suffices to assume that { κ \ B : B ∈ B} is anultrafilter and show that X is not normal.We consider X as a group with pointwise addition mod
2. For A ⊆ κ and α < κ , χ A denotes the characteristic function of A , [0 .α ] = { γ < κ : γ ≤ α } . Replacing κ to somecardinal κ ′ ≤ κ , we may suppose that [0 , α ] ∈ B for each α < κ .We consider two subsets Y, Z of X defined by Y = { y α : α < κ } , y α = χ { α } ,Z = { z α : α < κ } , z α = χ [0 ,α ] , IGOR PROTASOV, KSENIA PROTASOVA and show that
Y, Z are asymptotically disjoint. We take an arbitrary B ∈ B , denote by H the subgroup of X generated by { y α : α ∈ B } and take the minimal γ < κ such that γ ∈ κ \ B . Then ( y β + H ) ∩ Z = ∅ as soon as β ∈ κ \ B , β > γ .We suppose that X is normal and use Theorem 1.2 to choose a slowly oscillatingfunction f : X −→ [0 ,
1] such that f | Z ≡ f | Y ≡
0, and denote by U and V ultrafilterson X with the bases {{ y α : α ∈ κ \ B } : B ∈ B} , {{ z α : α ∈ κ \ B } : B ∈ B} . Let f β ( V + U ) = r , r ∈ [0 . . We take an arbitrary W ∈ V + U and pick C ∈ B and D α ∈ B , α ∈ C such that [ { z α + y β : α ∈ κ \ C, β ∈ κ \ D α } ⊆ W. Then we construct inductively two mappings ψ : κ \ C −→ κ, φ : κ \ C −→ κ such that ψ ( α ) ∈ κ \ D α , φ ( α ) ∈ κ \ D α and ψ ( κ \ C ) ∩ φ ( κ \ C ) = ∅ . Since { κ \ B : B ∈ B} isan ultrafilter, either ψ ( κ \ C ) ∈ B or φ ( κ \ C ) ∈ B . We assume that ψ ( κ \ C ) ∈ B anddenote by H the subgroup of X generated by { y α : α ∈ ψ ( κ \ C ) } . Then z α ∈ W + H for each α ∈ κ \ C . Hence, W and Z are linked. Since f is slowlyoscillating, we conclude that r = 1.On the other hand, for α ∈ κ \ C , z α + { y β : β ∈ κ \ D α } ⊆ W. It follows that W and Y are linked. Since f is slowly oscillating, we get r = 0 contradicting above paragraph. ✷ Corollary 4.2.
Let B be a bornology on a cardinal κ such that { κ \ B : B ∈ B} isan ultrafilter. Then the hyperballean [ κ ] B is not normal.Poof. Since a subballean of a normal ballean is normal, to apply Theorem 4.1, we useProposition 2.1. ✷ Theorem 4.1 answers Question 1 from [2], Corollary 4.2 answers Question 14.4 from [1].For a bornology B on a cardinal κ , the subballean of all characteristic functions offinite subsets of κ of the ballean { , } B is called the finitary B -macrocube. It followsfrom the proof of Theorem 4.1 that the finitary B -macrocube on ω is not normal providedthat { ω \ B : B ∈ B} is ultrafilter. Question 4.3.
Let B be a bornology on a cardinal κ such that { κ \ B : B ∈ B} isan ultrafilter. Is it true that the finitary B -macrocube is not normal? For n ∈ ω , n > B on κ , we denote [ κ ] B n = { A ∈ [ κ ] B : | A | ≤ n } , { , } B n = { ( x α ) α<κ : |{ α : x α = 1 }| ≤ n } . If { κ \ B : B ∈ B} is an ultrafilter then [ κ ] B n isnormal [1, Theorem 1.13].A ballean ( X, E ) is called ultranormal if any two unbounded subsets of X are linked. HE NORMALITY OF MACROCUBES AND HYPERBALLEANS 5
Theorem 4.4.
Let B be a bornology on κ such that { κ \ B : B ∈ B} is an ultrafilter.Then the ballean { , } B n is ultranormal.Proof. We consider { , } B as a subballean of { , } B n . Since { κ \ B : B ∈ B} is anultrafilter, it suffices to show that A , { , } B are linked for any unbounded subset A of { , } B n .Applying n -times Lemma 7.1 from [1], we pick P ⊆ κ such that κ \ P ∈ B and A ′ = { A ∈ A : | A ∩ P | = 1 } is unbounded. We put H = { ( x α ) α<κ : x α = 0 for each α ∈ P } . Then ( x + H ) ∩ A ′ = ∅ for each x ∈ P . Hence, A ′ and { , } B are linked. ✷ References [1] T. Banakh, I. Protasov,
The normality and bounded growth of balleans , https://arxiv.org/abs/1810.07979 .[2] T. Banakh, I. Protasov, Constructing balleans , J. Math. Sciences, (2019), 16-26.[3] T. Banakh, I. Zarichnyi,
Characterizing the Cantor bi-cube in asymptotic categories,
Groups Geom.Dyn. : 4 (2011), 691-728.[4] D. Dikranjan, I. Protasov, K. Protasova, N. Zava, Balleans, hyperballeans and ideals , Appl. Gen.Topology : 4 (2019), 431-447.[5] N. Hindman, D. Strauss, Algebra in the Stone- ˇ C ech Compactification , de Gructer, Berlin, New York,1998.[6] I. Protasov, Normal ball structures , Math. Stud. (2003), 3-16.[7] I. Protasov, T. Banakh, Ball Structures and Colorings of Groups and Graphs , Math. Stud. Monogr.Ser., Vol. 11, VNTL, Lviv, 2003.[8] I. Protasov, K. Protasova,
On hyperballeans of bounded geometry , Europ. J. Math. (2018), 1515-1520.[9] I. Protasov, M. Zarichnyi, General Asymptology , Math. Stud. Monogr. Ser., Vol. 12, VNTL, Lviv,2007, pp. 219.[10] J. Roe,
Lectures on Coarse Geometry , Univ. Lecture Ser., vol. 31, American Mathematical Society,Providence RI, 2003.
I.Protasov: Faculty of Computer Science and Cybernetics, Kyiv University, AcademicGlushkov pr. 4d, 03680 Kyiv, Ukraine
E-mail address : [email protected] K.Protasova: Faculty of Computer Science and Cybernetics, Kyiv University, AcademicGlushkov pr. 4d, 03680 Kyiv, Ukraine
E-mail address ::