aa r X i v : . [ m a t h . GN ] S e p MINMAX BORNOLOGIES
TARAS BANAKH, IGOR PROTASOV
Abstract.
A bornology B on a set X is called minmax if the smallest and the largest coarsestructures on X compatible with B coincide. We prove that B is minmax if and only if thefamily B ♯ = { p ∈ βX : { X \ B : B ∈ B} ⊂ p } consists of ultrafilters which are pairwisenon-isomorphic via B -preserving bijections of X . Also we construct a minmax bornology B on ω such that the set B ♯ is infinite. We deduce this result from the existence of a closedinfinite subset in βω that consists of pairwise non-isomorphic ultrafilters. Introduction
Let X be a set. A family E of subsets of X × X is called a coarse structure 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 ′ ) } and E − = { ( y, x ) : ( x, y ) ∈ E } ; • if E ∈ E and △ X ⊆ E ′ ⊆ E then E ′ ∈ E ; • S E = X × X .A subfamily E ′ ⊆ E is called a base for E if for every E ∈ E there exists E ′ ∈ E ′ suchthat E ⊆ E ′ . For x ∈ X , A ⊆ X and E ∈ E , we denote E [ x ] = { y ∈ X : ( x, y ) ∈ E } , E [ A ] = S a ∈ A E [ a ] and say that E [ x ] and E [ A ] are balls of radius E around x and A .The pair ( X, E ) is called a coarse space [9] or a ballean [7], [8].For a coarse space ( X, E ), a subset B ⊂ X is called bounded if B ⊂ E [ x ] for some E ∈ E and x ∈ X . A coarse space ( X, E ) is called unbounded if X is unbounded. In what follows,all balleans under consideration are supposed to be unbounded .A family B of subsets of an infinite set X is called a bornology on X if ∪B = X / ∈ B and B is closed under taking subsets and finite unions. For a coarse space ( X, E ), we denote by B ( X, E ) the bornology of all bounded subsets of ( X, E ).A coarse structure ( X, E ) is called • discrete if for every E ∈ E there exists B ∈ B ( X, E ) such that E [ x ] = { x } for each x ∈ X \ B ; • ultradiscrete if ( X, E ) is discrete and the family { X \ B : B ∈ B ( X, E ) } is an ultrafilter; • maximal if ( X, E ) is bounded in every strictly stronger coarse structure on X .Let B be a bornology on X . Following [2], we say that a coarse structure E on X is compatible with B if B = B ( X, E ) .By [2, § X , compatible with B , has the smallestand the largest elements ⇓B and ⇑B , respectively. Mathematics Subject Classification.
Key words and phrases.
Bornology, coarse structure, ballean, isomorphic and coherent ultrafilters.
The smallest coarse structure ⇓B , compatible with the bornology B , is generated by thebase consisting of the entourages ( B × B ) ∪ △ X , where B ∈ B . A coarse structure E on X isdiscrete if and only if ( X, B ) = ⇓B ( X, E ) . A discrete coarse structure is maximal if and only if E is ultradiscrete [8, Example 10.1.2].The largset coarse structure ⇑B , compatible with the bornology B , consists of all entourages E ⊂ X × X such that for any set B ∈ B the set E [ B ] ∪ E − [ B ] belongs to B , see [2, § Characterizing minmax bornologies
A bornology B on a set X is called minmax if ⇓B = ⇑B . Equivalently, B is minmax if B iscompatible with a unique coarse structure on X . It is clear that each ultradiscrete bornologyis minmax. In this section we show that the converse is not true.We recall that two ultrafilters p, q on a set X are isomorphic if there exists a bijection f : X → X such that the ultrafilter ¯ f ( p ) := { f ( P ) : P ∈ p } is equal to q .Let B be a bornology on a set X . We say that two ultrafilters p, q on X are B - isomorphic if there is a bijection f : X → X such that f ( p ) = q and { f ( B ) : B ∈ B} = B .We denote by B ♯ the set of all ultrafilters p on X such that { X \ B : B ∈ B} ⊂ p . Observethat a bornology B is ultradiscrete if and only if B ♯ is a singleton. Theorem 1.
A bornology B on a set X is minmax if and only if every two distinct ultrafilters p, q ∈ B ♯ are not B -isomorphic.Proof. To prove the “only if” part, assume that there exist two distinct B -isomorphic ul-trafilters p, q in B and take a bijection h : X → X witnessing this fact. Since p = q ,the set { x ∈ X : f ( x ) = x } does not belong to the bornology B . Then the entourage E = { ( x, y ) : x ∈ X : y ∈ { x, f ( x ) }} belongs to the coarse structure ⇑B and witnesses that itis not discrete and hence not equal to ⇓B . This means that B is not minmax.To prove the “if” part, assume that B is not minmax. Then the coarse structure ⇑B is notdiscrete and there exists E ∈ ⇑B such that the set { x ∈ X : | E [ x ] | > } does not belong tothe bornology B . We take a maximal by inclusion subset Y ⊆ { x ∈ X : | E [ x ] | > } such that E [ y ] ∩ E [ z ] = ∅ for all distinct y, z ∈ Y . We note that Y does not belong to B and and takean arbitrary ultrafilter p ∈ B ♯ such that Y ∈ p . For each y ∈ Y choose z y ∈ E [ y ] \ y andconsider the bijection f : X → X acting as the transposition on each pair y, z y and identicalon all other elements of X . Observe that { f ( B ) : B ∈ B} = B and hence p = ¯ f ( p ) ∈ B ♯ . So p and ¯ f ( p ) are two distinct B -isomorphic ultrafilters in B ♯ . (cid:3) The following example was first presented in [8, Example 1].
Example 1.
There exists a minmax bornology B on ω which is not ultradiscrete.Proof. Choose any two non-isomorphic ultrafilters p, q on ω and consider the bornology B = { B ⊂ ω : ω \ B ∈ p ∩ q } . Since B ♯ = { p, q } , the bornology B is minmax (by Theorem 1) andnot ultradiscrete. (cid:3) Now we shall construct a minmax bornology B on ω for which the set B ♯ has cardinality2 c . For this we need the following fact, which can have an independent value. Theorem 2.
The Stone- ˇCech compactification βω of ω contains a closed infinite subset con-sisting of pairwise non-isomorphic ultrafilters. INMAX BORNOLOGIES 3
Proof.
Let us recall that a point x of a topological space X is called a weak P -point if x doesnot belong to the closure of any countable subset C ⊂ X \ { x } . By Corollaries 4.5.2 and 4.3.2in [6], the space βω \ ω contains 2 c weak P -points. Consequently, we can choose a sequence offree ultrafilters ( p n ) n ∈ ω consisting of pairwise non-isomorphic weak P -points in βω \ ω . Thedefinition of a weak P -point implies that the subspace D = { p n } n ∈ ω of βω \ ω is discrete.Now the regularity of βω implies that there exists a family { P n } n ∈ ω of pairwise disjoint setsin ω such that P n ∈ p n for every n ∈ ω .We claim that the closure ¯ D of D consists of pairwise non-isomorphic ultrafilters. To derivea contradiction, assume that ¯ D contains two distinct isomorphic ultrafilters p, q . Then p ∈ ¯ P and q ∈ ¯ Q for some disjoint sets P, Q ⊂ D . Find a bijection f : ω → ω such that f ( p ) = q .The bijection f extends to a homeomorphism ¯ f : βω → βω . Since the set D consists ofpairwise non-isomorphic ultrafilters, ¯ f ( P ) is disjoint with Q . So, ¯ f ( P ) and Q are two disjointcountable discrete subspaces of βω \ ω consisting of weak P -points. Then ¯ f ( P ) is disjointwith the closure of Q and Q is disjoint with the closure of ¯ f ( P ) (which is equal to ¯ f ( ¯ P )).By Lemma 1 of Frol´ık [5] (see also Theorem 1.5.2 in [6]), ¯ Q ∩ ¯ f ( ¯ P ) = ∅ . On the other hand, q = ¯ f ( p ) ∈ ¯ Q ∩ ¯ f ( ¯ P ). (cid:3) Remark 1.
Theorem 2 has also a “near-coherent” version. Let us recall [3] that two ultra-filters p, q on ω are near-coherent if there exists a finite-to-one function f : ω → ω such that¯ f ( p ) = ¯ f ( q ). It is clear that any two isomorphic ultrafilters on ω are near-coherent. By [1], thespace βω contains an infinite closed set consisting of pairwise non-near-coherent ultrafiltersif and only if βω contains infinitely many non-near-coherent ultrafilters. The latter happensif u ≥ d . On the other hand, by [4, 9.18], the strict inequality u < g (which is consistent withZFC by [4, 11.2]) implies that all free ultrafilters on ω are near-coherent. Example 2.
There exist a minmax bornology B on ω such that |B ♯ | = 2 c .Proof. By Theorem 2, the space βω contains an infinite closed subset F consisting of pairwisenon-isomorphic ultrafilters. By Lemma 3.1.2(c) in [6], | F | = 2 c . Consider the bornology B = { B ⊂ ω : ω \ B ∈ T p ∈ F p } and observe that B ♯ = F . By Theorem 1, the bornology B isminmax. (cid:3) Characterizing bornologies with maximal coarse structure ⇑B By [8, Theorem 10.2.1], any unbounded set L in a maximal coarse space ( X, E ) is large (which means that X = E [ L ] for some E ∈ E ). The converse is not true: Example 3.
There exists a coarse structure E on a countable set X such that the coarsespace ( X, E ) is not maximal but each unbounded subset of ( X, E ) is large.Proof. Let X = ω , G be the group of all finitely supported permutations of X , and [ G ] <ω bethe family of all finite subsets of G . The action of the group G induces the coarse structure E = { E ⊂ X × X : ∃ F ∈ [ G ] <ω , △ X ⊂ E ⊂ { ( x, y ) : y ∈ { x } ∪ F x }} on X , whose bornology coincides with the bornology B of all finite subsets of X .The coarse structure E is not maximal since E ⊂ ⇑B and
E 6 = ⇑B . Indeed, the coarsestructure ⇑B contains the entourage E = S n ∈ ω [ n , ( n + 1) ) × [ n , ( n + 1) ) that does notbelong to the (finitary) coarse structure E .On the other hand, each unbounded set L ⊂ X is large since we can find a bijection f : X → X such that X \ L ⊂ f ( L ). This bijection determines the entourage E := { ( x, y ) ∈ X × X : y ∈ { x, f ( x ) }} ∈ E such that E [ L ] = X . (cid:3) TARAS BANAKH, IGOR PROTASOV
Theorem 3.
For a bornology B on a set X , the coarse space ( X, ⇑B ) is maximal if and onlyif each unbounded subset of ( X, ⇑B ) is large.Proof. The “only if” part follows from Theorem 10.2.1 in [8].To prove the “if” part, assume that each unbounded subset of ( X, ⇑B ) is large, but ( X, ⇑B )is not maximal. Then there is an unbounded coarse structure E on X such that ⇑B ( E .By the definition of ⇑B , the coarse structure E is not compatible with the bornology B .Consequently, there exists a set L ⊂ X which is bounded in ( X, E ) but does not belong to B .Then L is unbounded in ( X, ⇑B ) and hence X = E [ L ] for some E ∈ ⇑B ⊂ E , which impliesthat X is bounded in ( X, E ). But this contradicts the choice of E . (cid:3) Example 4.
Theorem 3 implies that for any infinite set X and the bornology B := { A ⊂ X : | A | < | X |} the coarse structure ⇑B is maximal. Indeed, for any subset L ⊂ X of cardinality | L | = | X | , we can find a bijection f of X such that X \ L ⊂ f ( L ). Then E = { ( x, y ) ∈ X × X : y ∈ { x, f ( x ) }} is an entourage in ⇑B such that X = E [ L ], which means that the set L in large in ( X, ⇑B ).Following [2], we say that a coarse structure E on X is relatively maximal if E = ⇑B ( X, E ) .Clearly, E is relatively maximal if either E is maximal or B ( X, E ) is minmax. Question 1.
Given a coarse structure E , how can one detect whether E is relatively maximal? Remark 2.
In light of Theorem 1, it is a very rare case when the coarse structure E on X isuniquely defined by the bornology B ( X, E ) .We denote by so ( X, E ) the set of all slowly oscillating functions of ( X, E ). By Theorem 7.3.1[8], if the coarse structures E and E ′ on X have linearly ordered bases and B ( X, E ) = B ( X, E ) ′ , so ( X, E ) = so ( X, E ′ ), then E = E ′ . We denote by δ ( X, E ) the binary relation on the power-set2 X of X defined by Aδ ( X, E ) B if and only if there exists E ∈ E such that A ⊆ E [ B ] and B ⊆ E [ A ]. By Theorem 7.5.3 from [8], if coarse structures E , E ′ on X have linearly orderedbases and δ ( X, E ) = δ ( X, E ′ ) then E = E ′ . INMAX BORNOLOGIES 5
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E-mail address : [email protected] I.Protasov: Faculty of Computer Science and Cybernetics, Kyiv University, Academic Glushkovpr. 4d, 03680, Kyiv, Ukraine
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