
TL;DR
This paper explores extremal properties of balleans, introducing concepts like ultranormal and extremely normal, and constructs examples illustrating these extremal conditions within coarse geometry.
Contribution
It defines and investigates the properties of ultranormal and extremely normal balleans, establishing their relationships and providing concrete examples with extremal characteristics.
Findings
Every maximal ballean is extremely normal.
Every extremely normal ballean is ultranormal.
A discrete ballean is ultranormal iff it is maximal.
Abstract
A ballean (or coarse space) is a set endowed with a coarse structure. A ballean is called normal if any two asymptotically disjoint subsets of are asymptotically separated. We say that a ballean is ultranormal (extremely normal) if any two unbounded subsets of are not asymptotically disjoint (every unbounded subset of is large). Every maximal ballean is extremely normal and every extremely normal ballean is ultranormal, but the converse statements do not hold. A normal ballean is ultranormal if and only if the Higsons corona of is a singleton. A discrete ballean is ultranormal if and only if is maximal. We construct a series of concrete balleans with extremal properties.
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Taxonomy
TopicsAdvanced Topology and Set Theory · Mathematical Dynamics and Fractals · Advanced Banach Space Theory
Extremal balleans
Igor Protasov
Abstract. A ballean (or coarse space) is a set endowed with a coarse structure. A ballean is called normal if any two asymptotically disjoint subsets of are asymptotically separated. We say that a ballean is ultranormal (extremely normal) if any two unbounded subsets of are not asymptotically disjoint (every unbounded subset of is large). Every maximal ballean is extremely normal and every extremely normal ballean is ultranormal, but the converse statements do not hold. A normal ballean is ultranormal if and only if the Higson*′*s corona of is a singleton. A discrete ballean is ultranormal if and only if is maximal. We construct a series of concrete balleans with extremal properties.
**MSC: ** 54E35.
Keywords: Ballean, coarse structure, bornology, maximal ballean, ultranormal ballean, extremely normal ballean.
1. Introduction
Let be a set. A family of subsets of is called a *coarse structure * if
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each contains the diagonal , ;
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if , then and , where , ;
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if and then ;
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for any , there exists such that .
A subset is called a base for if, for every , there exists such that . For , and , we denote , and say that and are balls of radius around and .
The pair is called a coarse space [14] or a ballean [10], [12].
Each subset defines the subballean , where is the restriction of to . A subset is called bounded if for some and .
Given a ballean , a subset of is called
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large if there exists such that ;
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small if is large for each large subset ;
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thick if, for every , there exists such that .
Every metric on a set defines the *metric ballean * , where has the base . A ballean is called metrizable if there exists a metric on such that . A ballean is metrizable if and only if has a countable base [12, Theorem 2.1.1]. Let be a ballean. A subset of is called an asymptotic neighbourhood of a subset if, for every , is bounded.
Two subsets of are called
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asymptotically disjoint if, for every , is bounded;
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asymptotically separated if have disjoint asymptotic neighbourhoods.
A ballean is called normal [7] if any two asymptotically disjoint subsets are asymptotically separated. Every ballean with linearly ordered base of , in particular, every metrizable ballean is normal [7, Proposition 1.1].
A function is called slowly oscillating if, for any and , there exists a bounded subset of such that *diam * for each . By [7, Theorem 2.2], a ballean is normal if and only if, for any two disjoint and asymptotically disjoint subsets of , there exists a slowly oscillating function such that and .
We say that an unbounded ballean is
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*ultranormal * [1] if any two unbounded subsets of are not asymptotically disjoint;
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extremely normal if any unbounded subset of is large.
An unbounded ballean , is called maximal if is bounded in any stronger coarse structure. By [13], every unbounded subset of a maximal ballean is large and every small subset is bounded. Hence, every maximal ballean is extremely normal.
A family of subsets of a set , closed under finite unions and subsets, is called a bornology if . For every ballean , the family of all bounded subsets of is a bornology.
A ballean is called discrete (or pseudodiscrete [12], or thin [5]) if, for every , there exists such that for each . Every bornology on a set defines the discrete ballean , where if and if . It follows that every discrete ballean is uniquely determined by its bornology , see [12, Theorem 3.2.1]. Every discrete ballean is normal [12, Example 4.2.2].
For a ballean the following conditions are equivalent: is discrete, every function is slowly oscillating, every unbounded subset of is thick, see [12, Theorem 3.3.1] and [3, Theorem 2.2].
An unbounded discrete ballean is called *ultradiscrete * if the family is an ultrafilter on . Every ultradiscrete ballean is maximal [12, Example 10.1.2]. Thus, we have got
ultradiscrete maximal extremely normal ultranormal
and no one arrow can be reversed, see Section 3.
2. Characterizations
Let be an unbounded ballean. We endow with the discrete topology, identify the Stone-ech compactification of with the set of all ultrafilters on and denote is unbounded for each . Given any , we write if there exists such that for each . By [7, Lemma 4.1], is an equivalence relation on . We denote by the minimal (by inclusion) closed (in ) equivalence on such that . The compact Hausdorff space is called the corona of , it is denoted by . If every ball in the metric space is compact then is the Higson’s corona of , see [8] and [14].
We say that a function is constant at infinity if there exists such that, for each , the set is bounded. We say that is almost constant if const for some . If the bornology is closed under countable unions then every function, constant at infinity, is almost constant.
If is constant at infinity then is slowly oscillating. We denote so the set of all bounded slowly oscillating functions on . For a bounded function , denotes the extension of to .
Theorem 1. For an unbounded normal ballean , the following conditions are equivalent:
(1)\* every function is constant at infinity;*
(2)\* is a singleton;*
(3)\* is ultranormal.*
Proof. . We assume that a function is not constant at infinity. Then there exists distinct and such that f^{\beta}(p)=a,\ f^{\beta}(q)=b.\ We put and choose , such that , . Given an arbitrary , we take such that for each . It follows that so and are asymptotically disjoint and we get a contradiction to (3).
. Let . By Proposition 8.1.4 from [12], if and only if for every .
. We assume that is not ultranormal and choose two disjoint and asymptotically disjoint unbounded subsets of . We chose such that , . Since is normal, there exists such that , . Then and by Proposition 8.1.4 from [12].
An unbounded ballean is called irresolvable [11] if can not be partitioned into two large subsets.
Theorem 2. For an unbounded discrete ballean , the following conditions are equivalent:
(1)\* is ultradiscrete;*
(2)\* is extremely normal;*
(3)\* is ultranormal;*
(4)\* is maximal and irresolvable.*
Proof. . We denote by the ultrafilter . If is an unbounded subset of then so is bounded and is large.
. Let be unbounded subsets of . Since is large, there exists such that so and are not asymptotically disjoint.
. If is not ultradiscrete then there exist two disjoint unbounded subsets of . Since is discrete, and are asymptotically disjoint.
. This is Theorem 10.4.5 from [12].
Let be a bornology on a set . Following [1], we say that a coarse structure is compatible with if is the bornology of all bounded subsets of .
Each bornology on defines two coarse structures and , the smallest and the largest coarse structures on compatible with . Clearly, is the discrete coarse structure defined by , in particular, is normal.
The coarse structure consists of all entourages such that and for each . But in contrast to , the coarse structure needs not to be normal [2, Theorem 12].
If a ballean is maximal then .. It follows that every maximal ballean is uniquely determined by the bornology of bounded subsets: if is maximal and then .
Proposition 1. Let be an extremely normal ballean and let be coarse structure on such that and is maximal. Then and .
Proof. We assume the contrary and pick . Since is extremely normal, is large in . Hence, is large in so is bounded and we get a contradiction to maximality of .
Following [2], we say that a ballean is relatively maximal if .
We recall that two ultrafilters are incomparable if, for every , we have , .
Proposition 2. Let be an unbounded discrete ballean such that any two distinct utrafilters from are incomparable. Then is relatively maximal.
Proof. We suppose the contrary and choose a coarse structure on such that and . Then there exists such that the set is unbounded in . For each , we pick , and extend to by for each . We take an arbitrary ultrafilter such that . By the assumption, is bounded in for some , . Then must be bounded in and we get a contradiction with the choice of .
3. Constructions
Given a family of subsets of , we denote by the intersections of all coarse structures, containing each , and say that is generated by . It is easy to see that has a base of subsets of the form , where
[TABLE]
If and are coarse structures, we write in place of . For lattices of coarse structures, see [11].
Proposition 3. Let be an unbounded discrete ballean and let be distinct ultrafilters from . If is relatively maximal then for each such that, for every , and .
Proof. We assume the contrary and choose such that . We put and denote by the coarse structure generated by and . Since is discrete and is not discrete, we have .
We take and . Applying an induction by and the assumptions , for each , we conclude that . Hence, and we get a contradiction to relative maximality of .
Proposition 4. Every unbounded subballean of maximal (extremely normal, ultranormal) ballean is maximal (extremely normal, ultranormal).
Proof. We prove only the first statement, the second and third are evident.
Let be a maximal ballean, be an unbounded subset of . We assume that is not maximal and choose a coarse structure on such that and is not bounded. We put . Since is maximal and , must be bounded. On the other hand, each bounded subset of is bounded in because otherwise is large in so is large in . Now let , and On induction by , we see that is bounded in . Hence is not bounded and we get a contradiction.
Proposition 5. Let be coarse structures on a set such that . Then the following statements hold:
(1)\ \* if is extremely normal then is extremely normal;*
(2)\ \* if is ultranormal and then is ultranormal;*
Proof. (1)\ \ Let be a subset of . If is unbounded in then is unbounded in . If is large in then is large in .
(2)\ \ We assume that some unbounded subsets of are asymptotically disjoint in . Then for each . Since and , we have for each so are asymptotically disjoint in .
Example 1. For every infinite regular cardinal , we construct a coarse structure on such that is maximal and . We denote by the family of all coverings of defined by the rule: if and only if, for each and , and , . Then is defined by the base , where M_{\mathcal{P}}=\{(x,y):x\in P,y\in P\ for some . For general construction of coarse structures by means of coverings, see [9] or [12, Section 7.5]. Clearly, and is maximal [12, Example 10.2.1].
Let be a group and let be a -space with the action , . A bornology on is called a group bornology if, for any , we have , . Every group bornology on defines a coarse structure on with the base , where is the identity of , . Moreover, every coarse structure on can be defined in this way [6].
Example 2. We define a coarse structure on such that the ballean is extremely normal but is not maximal. Let denotes the group of all permutations of , , . We show that every infinite subset of is large. We partition and into two infinite subset , and choose two permutations of so that
, , \ \ f_{1}(x)=x\ \ for each ,
, , \ \ f_{2}(x)=x\ \ for each
Then we put , where is the identity permutation. Clearly, so is extremely normal. To see that is not maximal, we note that and choose a partition of such that . Then, for each , there exist such that . For each and , we have . Hence, .
Example 3. Let be a cardinal, . We construct two coarse structures on such that , is ultranormal but not extremely normal, is extremely normal but not maximal. Let denotes the group of all permutations , , . Clearly, . Let be infinite subsets of . We take countable subset , and use argument from Example 2 to choose so that . It follows that are not asymptotically disjoint in so is ultranormal. If is a countable subset of then for each so is not extremely normal.
We denote by the discrete coarse structure on defined by the bornology and put . If is an unbounded subset of then . Applying the arguments from Example 2, we see that is extremely normal. The ballean is not maximal because , where .
4. Comments
- Let be a ballean, and be subsets of . Following [9], we write if and only if there exists such that , .
Let be coarse structures on a set such that . If and have linearly ordered bases and either or then , see [4, Theorem 2.1] and [3, Theorem 4.2].
We take the coarse structures and on from Example 2. By Theorem 1, so . Since and are extremely normal, we have . By the construction, .
In this connection we remind Question 7.5.1 from [12]: does imply , and give the affirmative answer to this question.
Clearly, implies . Assume that there exists . Then there exist and such that, for each , one can find , such that but We put and choose a function such that , for each . Let and , be an ultrafilter with the base . Then and there exists such that for all , . Since , and are not close in but , are close in .
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Every group has the natural finitary coarse structure with the base , where . Let be an uncountable abelian group. By [4, Corollary 3.2], is not normal but every function is constant at infinity. This example shows that the assumption of normality in Theorem 1 can not be omitted.
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Example 3 shows that Proposition 1 does not hold for ultranormal ballean in place of extremely normal. Indeed, is ultranormal, , , . We take a maximal coarse structure such that . Then . and .
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Following [1], we say that a ballean *has bounded growth * if there is a mapping such that
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for each ;
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for each , there exists such that for each .
Clearly, every discrete ballean has bounded growth . Let us take the maximal ballean from Example 1. Since each ball in is finite, one can use the diagonal process to show that is not of bounded growth.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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- 5[5] Ie. Lutsenko, I. Protasov, Thin subsets of balleans , App. Gen. Topol. 11 (2010), 89-93.
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