Ultrafilters on balleans
A ballean (equivalently, a coarse structure) is an asymptotic counterpart of a uniform space. We introduce three ultrafilter satellites of a ballean (namely, corona, companion, and corona companion), evaluate the basic cardinal invariants of the corona and characterize the subsets of balleans in ter...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508032812711936 |
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| author | Protasov, I. V. Slobodianiuk, S. V. Протасов, І. В. Слободянюк, С. В. |
| author_facet | Protasov, I. V. Slobodianiuk, S. V. Протасов, І. В. Слободянюк, С. В. |
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| description | A ballean (equivalently, a coarse structure) is an asymptotic counterpart of a uniform space. We introduce three ultrafilter satellites of a ballean (namely, corona, companion, and corona companion), evaluate the basic cardinal invariants of the corona and characterize the subsets of balleans in terms of companions. |
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UDC 512.5
I. V. Protasov, S. V. Slobodianiuk (Kyiv Nat. Taras Shevchenko Univ.)
ULTRAFILTERS ON BALLEANS
УЛЬТРАФIЛЬТРИ НА БОЛЕАНАХ
A ballean (equivalently, a coarse structure) is an asymptotic counterpart of a uniform space. We introduce three ultrafilter
satellites of a ballean (namely, corona, companion, and corona companion), evaluate the basic cardinal invariants of the
corona and characterize the subsets of balleans in terms of companions.
Болеан (або груба структура) — це асимптотичний аналог рiвномiрного простору. За допомогою ультрафiльтрiв
визначено три супутники болеанiв (а саме, корону, компаньйон i коронний компаньйон), знайдено оцiнки основних
кардальних iнварiантiв корони та охарактеризовано пiдмножини болеанiв за допомогою компаньйонiв.
1. Introduction. A ball structure is a triple B = (X,P,B), where X, P are nonempty sets, B :
X × P → PX , x ∈ B(X,α) for each x ∈ X and α ∈ P, PX denotes the family of all subsets of
X. The set X is called the support of B, P is called the set of radii and B(x, α) is called a ball of
radius α around x.
Given any x ∈ X, A ⊆ X, α ∈ P, we set
B∗(x, α) = {y ∈ X : x ∈ B(y, α)}, B(A,α) =
⋃
a∈A
B(a, α).
A ball structure B = (X,P,B) is called a ballean if
for any α, β ∈ P, there exist α′, β′ such that, for every x ∈ X,
B(x, α) ⊆ B∗(x, α′), B∗(x, β) ⊆ B(x, β′);
for any α, β ∈ P, there exists γ ∈ P such that, for every x ∈ X,
B(B(x, α), β) ⊆ B(x, γ);
for any x, y ∈ X, there exists α ∈ P such that y ∈ B(x, α).
A ballean B on X can also be defined in terms of entourages of the diagonal ∆X of X ×X (in
this case it is called a coarse structure [1]), and can be considered as an asymptotic counterpart of a
uniform space. For our goals, we prefer the ball language from [2, 3].
Let B = (X,P,B), B′ = (X ′, P ′, B′) be balleans. A mapping f : X → X ′ is called a≺-mapping
if, for every α ∈ P, there exists α′ ∈ P ′ such that, for every x ∈ X, f(B(x, α)) ⊆ B′(f(x), α′).
If there exists a bijection f : X → X ′ such that f and f−1 are ≺-mappings, B and B′ are called
asymorphic and f is called an asymorphism.
For a ballean B = (X,P,B), a subset Y ⊆ X is called large if there is α ∈ P such that
X = B(Y, α). A subset V of X is called bounded if V ⊆ B(x, α) for some x ∈ X and α ∈ P. Each
nonempty subset Y ⊆ X determines a subballean BY = (Y, P,BY ), where BY (y, α) = Y ∩B(y, α).
We say that B and B′ are coarsely equivalent if there exist large subset Y ⊆ X and Y ′ ⊆ X ′
such that the subballeans BY and B′Y ′ are asymorphic.
Given a ballean B = (X,P,B), x, y ∈ X and α ∈ P, we say that x and y are α-path connected if
there exists a finite sequence x0, . . . , xn, x0 = x, xn = y such that xi+1 ∈ B(xi, α), xi ∈ B(xi+1, α)
c© I. V. PROTASOV, S. V. SLOBODIANIUK, 2015
1698 ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12
ULTRAFILTERS ON BALLEANS 1699
for each i ∈ {0, . . . , n− 1}. For any x ∈ X and α ∈ P, we denote
B�(x, α) = {y ∈ X : x, y are α-path connected}.
The ballean B� = (X,P,B�) is called a cellularization of B. A ballean B is called cellular if the
identity mapping id : X → X is an asymorphism between B and B�. By [3] (Theorem 3.1.3), B is
cellular if and only if B is asymptotically zero-dimensional.
For a ballean B = (X,P,B), we use a natural preordering on P defined by the rule: α < β if
and only if B(x, α) ⊆ B(x, β) for each x ∈ X. A subset P ′ ⊆ P is called cofinal if for every α ∈ P,
there is α′ ∈ P ′ such that α < α′. The minimal cardinality cfB of cofinal subsets of P is called
cofinality of B.
A ballean B is called ordinal if there exists a cofinal subset of P well-ordered by < . Up to
asymorphism, we can replace P with some segment [0, γ) of ordinals and, moreover, we can assume
that γ is a regular cardinal. It is easy to see that every ordinal ballean of uncountable cofinality is
cellular. More on cellular balleans can be found in [3] (Chapter 3).
Let B = (X,P,B) be a ballean. We say that two subsets Y, Z of X are asymptotically disjoint if,
for every α ∈ P, there exists a bounded subset Vα of X such that B(Y \Vα, α)∩B(Z \Vα, α) = ∅.
The subsets Y, Z are called asymptotically separated if to each α ∈ P one can assign a bounded
subset Vα of X such that( ⋃
α∈P
B(Y \ Vα, α)
)⋂( ⋃
α∈P
B(Z \ Vα, α)
)
= ∅.
A ballean B is called normal if any two asymptotically disjoint subsets of X are asymptotically
separated. For normal balleans see [4] and [3] (Chapter 4). According to [3] (Chapter 4), every
ordinal ballean is normal.
In Section 2 we give some examples of cellular and ordinal balleans. In Section 3 we introduce
three ultrafilter satellites of a ballean: corona, ultracompanion and corona companion. In Section 4
we evaluate the basic cardinal invariants of coronas of ordinal balleans. In Section 5 we characterize
the subsets of a ballean in terms of its ultracompanions and corona companions.
2. Examples.
Example 2.1. Each metric space (X, d) defines a metric ballean (X,R+, Bd), where Bd(x, r) =
= {y ∈ X : d(x, y) ≤ r}. By [3] (Theorem 2.1.1), for a ballean B, the following conditions are
equivalent:
B is asymorphic to some metric ballean;
B is coarsely equivalent to some metric ballean;
cfB ≤ ℵ0.
Clearly, each metric ballean is ordinal. By [3] (Theorem 3.1.1), a metric ballean B is cellular if
and only if B is asymorphic to a ballean of some ultrametric space.
Example 2.2. Every infinite cardinal κ defines the cardinal ballean←→κ = (κ, κ,
←→
B ), where
←→
B (x, α) = {y ∈ κ : x ≤ y ≤ x+ α or y ≤ x ≤ y + α}.
For cardinal ballean see [5]. In particular [5] (Theorem 3), if κ > ℵ0 then ←→κ is cellular. Clearly,
each cardinal ballean is ordinal.
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1700 I. V. PROTASOV, S. V. SLOBODIANIUK
Example 2.3. Let γ be a limit ordinal, {µα : α < γ} be a family of cardinals. A direct product
⊗α<γµα is a set of all γ-sequences x = (xα)α<γ such that xα ∈ µα and xα = 0 for all but finitely
many α < γ. We consider a ballean
B = (⊗α<γµα, [0, γ), B),
where B(x, β) = {y ∈ ⊗α<γµα : yα = xα for every α ≥ β}. Evidently, B is ordinal and cellular.
For decompositions of balleans into direct products see [6] and [7].
Example 2.4. Let G be a group. An ideal J in the Boolean algebra PG of all subsets of G is
called a group ideal if J contains all finite subsets of G and if A,B ∈ J then AB−1 ∈ J.
Now let X be a transitive G-space with the action G × X → X, (g, x) 7→ gx, and let J be a
group ideal in G. We define a ballean B(G,X, J) as a triple (X, J,B), where B(x,A) = Ax ∪ {x}
for all x ∈ X, A ∈ J. By [8] (Theorem 1), every ballean B with the support X is asymorphic to the
ballean B(G,X, J) for some group G of permutations of X and some ideal J of G. By [8] (Theorem
3), every cellular ballean B with the support X is asymorphic to B(G,X, J) for some group G of
permutations of X and some ideal J which has a base consisting of subgroups.
In the case X = G, and the left regular action of G on X, we write (G, J) instead B(G,X, J).
3. Ultrafilters. Let B = (X,P,B) be an unbounded ballean. We endow X with the discrete
topology and consider the Stone – Čech compactification βX ofX.We take the points of βX to be the
ultrafilters on X with the points of X identified with the principal ultrafilters on X. For every subset
A ⊆ X, we put A = {q ∈ βX : A ∈ q}. The topology of βX can be defined by stating that the family
{A : A ⊆ X} is a base for the open sets. Let Y be a compact Hausdorff space. For a mapping f :
X → Y, fβ denotes the Stone – Čech extension of f onto βX.
We denote by X]
B the set of all ultrafilters on X whose members are unbounded in B, and note
that X]
B is a closed subset of βX.
Given any r, q ∈ X]
B, we say that r, q are parallel (and write r||q) if there exists α ∈ P such
that B(R,α) ∈ q for every R ∈ r. By [4] (Lemma 4.1), || is an equivalence on X]
B. We denote by ∼
the minimal (by inclusion) closed (in X]
B ×X
]
B) equivalence on X] such that || ⊆∼ . The quotient
X]
B/ ∼ is a compact Hausdorff space. It is called the corona of B and is denoted by X̌B. Let (X, d)
be a metric space such that each closed ball in X is compact, B = B(X, d). Then X̌B coincides with
the Higson’s corona of (X, d) (see [9, p. 154]).
For every p ∈ X]
B, we denote by p̌ the class of the equivalence ∼, and say that two ultrafilters
p, q ∈ X]
B are corona equivalent if p̌ = q̌. To detect whether two ultrafilters p, q ∈ X]
B are corona
equivalent we use the slowly oscillation functions.
A function h : X → [0, 1] is called B-slowly oscillating if, for every ε > 0 and every α ∈ P,
there exists a bounded subset V of X such that
diamh(B(x, α)) < ε
for each x ∈ X \ V.
Proposition 3.1. Let B = (X,P,B) be an unbounded ballean, q, r ∈ X]
B. Then q̌ = ř if and
only if hβ(p) = hβ(q) for every B-slowly oscillating function h : X → [0, 1].
Proof. See [9] (Proposition 1).
Proposition 3.2. Let B = (X,P,B) be an unbounded normal ballean, q, r ∈ X]
B. Then q̌ = ř if
and only if for any Q ∈ q and R ∈ r, there exists α ∈ P such that B(Q,α)∩B(R,α) is unbounded.
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ULTRAFILTERS ON BALLEANS 1701
Proof. See [4] (Lemma 4.2).
Proposition 3.3. Let B = (X,P,B) be an unbounded normal ballean and let q ∈ X]. Then the
family of subsets of the form ř ∈ X̌B :
⋃
α∈Q
B(Q \ Vα, α) ∈ r
,
where Q ∈ q and each subset Vα is bounded, is a base of the neighborhoods of the point q̌ in X̌B.
Proof. See [4, p. 15].
Proposition 3.4. Let B = (X,P,B) be a ballean, Y ⊆ X, α ∈ P, q ∈ X]. If B(Y, α) ∈ q, then
there is r ∈ Y ] such that q||r.
Proof. For each Q ∈ q, we denote SQ = B(Q,α) ∩ Y and note that the family {SQ : Q ∈ q} is
contained in some ultrafilter r ∈ Y ]. Clearly, r||q.
Proposition 3.4 is proved.
We note that, for a cellular ballean B = (X,P,B), corona X̌B coincides with its binary corona
(see [3], Chapter 8) and hence X̌B is zero-dimensional.
Let B = (X,P,B) be a ballean, A ⊆ X, p ∈ X] and ¯̄p = {q ∈ X] : p||q}. A subset
∆p(A) = ¯̄p ∩A]
is called an ultracompanion of A. For ultracompanions of subsets of metric spaces, groups and
G-spaces see [10 – 13].
Given a ballean B = (X,P,B) and a subset A of X, we say that the subset p̌ ∩ A] is a corona
companion of A.
4. Cardinal invariants. Given a ballean B = (X,P,B), a subset A of X is called
large if X = B(A,α) for some α ∈ P ;
small if X \B(A,α) is large for every α ∈ P ;
thick if, for every α ∈ P, there exists a ∈ A such that B(a, α) ⊆ A;
thin if, for every α ∈ P, there exists a bounded subset V of X such that B(a, α)∩B(a′, α) = ∅
for all distinct a, a′ ∈ A \ V.
We note that large, small, thick and thin subsets can be considered as asymptotic counterparts
of dense, nowhere dense, open and discrete subsets of a uniform topological space. We use the
following cardinal invariants of B : asymptotic density, thickness and spread defined by
asdenB = min{|L| : L is a large subset of X},
thickB = sup{|F| : F is a family of pairwise disjoint thick subsets of X},
spreadB = sup{|Y |B : Y is a thin subset of X}, where |Y |B = min{|Y \ V | : V is a bounded
subset of X}.
Theorem 4.1. For every unbounded ordinal ballean B with the support X, we have
asdenB = thickB = spreadB,
and there exists a thin subset Y of X and a disjoint family F of thick subsets of X such that
|Y |B = |Y | = asdenB = |F|.
Proof. See [14] (Theorem 3.1) and [15] (Theorem 2.3).
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12
1702 I. V. PROTASOV, S. V. SLOBODIANIUK
Theorem 4.2. Let B = (X,P,B) be an unbounded ordinal ballean and let κ = asdensB. Then
|X̌B| = 22
κ
.
Proof. Let Z be a large subset of X such that |Z| = κ. By Proposition 3.4, p̌∩Z] 6= ∅ for each
p ∈ X]. Hence, |XB| ≤ |Z]| ≤ βZ = 22
κ
.
To verify the inequality |X̌B| ≥ 22
κ
, we use Theorem 4.1 to find a thin subset Y of X such that
|Y |B = |Y | = κ. Since Y is thin and B is normal, by Proposition 3.2, p̌ 6= q̌ for any two distinct
ultrafilters p, q from Y ]. So it suffices to prove that |Y ]| = 22
κ
. We fix some y0 ∈ Y and consider
two cases.
Case 1. There exists a cofinal subset C = {cα : α < λ} in P such that |Y ∩ (B(y0, cα+1) \
B(y0, cα))| and choose some ultrafilter p on C such that {cβ : α < β < λ} ∈ p for each α < λ. If
q, q′ ∈ βκ and q 6= q′ then
p-limfα(q) 6= p-limfα(q′),
so |Y ]| ≥ βκ = 22
κ
.
Case 2. There exists β ∈ P such that |Y ∩ (B(y0, α) \B(y0, β))| < κ for each α > β. We put
Z = Y \B(y0, β) and note that |Z| = κ. By the choice of β, Z] coincides with the set of all uniform
ultrafilters on Z so |Z]| = 22
κ
.
Theorem 4.2 is proved.
Let κ be an infinite cardinal, φ be a uniform ultrafilter on κ. We consider a ballean B = (κ, φ,B),
where, for any F ∈ φ, B(x, F ) = {x} if x ∈ F and B(x, F ) = X \F if x /∈ F. Clearly, B is normal
but κ] = {φ} so κ̌B is a singleton. On the other hand asdensB = κ. So Theorem 4.2 does not hold
for B.
Recall that the Souslin number s(X) of a topological space X is the supremum of cardinalities
of disjoint families of open subsets of X.
Let κ be an infinite cardinal. A family F of subsets of κ is called almost disjoint if |F | = κ for
every F ∈ F , and |F ′ ∩ F | < κ for all distinct F, F ′ ∈ F . For κ = ℵ0, there is an almost disjoint
family of cardinality c. Baumgartner [16] proved that, for each κ, there is an almost disjoint family of
cardinality κ+, and it is independent of ZFC that if κ = ℵ1 then there is no almost disjoint families
of cardinality 2κ.
Proposition 4.1. Let κ be an infinite cardinal, B = (X,κ, P ) be an unbounded ordinal ballean.
Assume that there exists a subset Y = {yα : α < κ} of X such that
B(B(yα, α), α) ∩B(B(yβ, β), β) = ∅
for all α < β < κ. If F is an almost disjoint family of subsets of κ, then
s(X̌B) ≥ |F|.
Proof. For each F ∈ F , we put
YF =
⋃
α∈F
B(yα, α), ZF = {q̌ : YF ∈ q}.
Applying Propositions 3.2 and 3.3 we conclude that {ZF : F ∈ F} is a disjoint family of subsets of
X̌B and each ZF has a nonempty interior.
Proposition 4.1 is proved.
Recall that the density denX of a topological space X is the smallest cardinality of dense subset
of X.
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ULTRAFILTERS ON BALLEANS 1703
Proposition 4.2. For every unbounded ballean B = (X,P,B), we have
den(X̌B) ≤ 2asdenX .
Proof. We take a large subset L of X of cardinality asdenX and denote by F the family of
all unbounded subsets of L. For each F ∈ F , we pick qF ∈ X] such that F ∈ qF . Then {q̌F :
F ∈ F} is a dense subset of X̌B and |{q̌F : F ∈ F}| ≤ 2|L|.
Proposition 4.2 is proved.
Theorem 4.3. For every infinite cardinal κ, we have asden←→κ = κ and
κ+ ≤ (κ̌) ≤ den(κ̌) ≤ 2κ,
where κ̌ is the corona of←→κ .
Proof. By the definition of κ, each large subset of κ has cardinality κ. By Proposition 4.2,
den(κ̌) ≤ 2κ. In view of Proposition 4.1 and the Baumgartner theorem it suffices to construct
corresponding subset Y.
We put y0 = 1 and define a set Y = {yα : α < κ} recursively by yα+1 = yα + yα + yα + yα and
yβ = sup{yα : α < β} for every limit ordinal β.
Theorem 4.3 is proved.
For κ = ℵ0, Proposition 4.1 gives more strong result
s(ℵ̌0) = den(ℵ̌0) = c.
Recall that a character χ(x) of a topological space X at the point x is the minimal cardinality of
bases of neighborhoods of x.
For a metric space X, X̌ denotes the corona of corresponding metric ballean. Under CH, if X
is a countable ultrametric space then, by [17], X̌ is homeomorphic to ω∗ = βω \ ω. By [18], this
statement is independent of ZFC.
Theorem 4.4. Let X be an unbounded metric space, κ = asden X̌. Then |X̌| = 22
κ
and
c · κ ≤ s(X̌) ≤ den(X̌) ≤ 2κ.
Proof. In view of Theorem 4.2 and Proposition 4.2, it suffices to verify only c · κ ≤ s(X̌).
The inequality c ≤ s(X̌) follows directly from Proposition 4.1. To prove κ ≤ s(X̌), we use
Theorem 3.1 and choose a disjoint family F of thick subsets of X such that |F| = κ. For each
F ∈ F , use Proposition 3.3 to find a subset XF ⊆ F such that projections p 7→ p̌ of {X]
F :
F ∈ F} to corona are pairwise disjoint with nonempty interior.
Theorem 4.4 is proved.
Corollary 4.1. For an unbounded countable metric space X we have
den(X̌) = s(X̌) = c.
5. Companions.
Theorem 5.1. Let B be a ballean with the support X. For a subset A of X, the following
statements hold:
(i) A is large if and only if ∆p(A) 6= ∅ for each p ∈ X];
(ii) A is thick if and only if ¯̄p = ∆p(A) for some p ∈ X];
(iii) A is prethick if and only if there exist p ∈ X] and α ∈ P such that ¯̄p = ∆p(B(A,α));
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12
1704 I. V. PROTASOV, S. V. SLOBODIANIUK
(iv) A is thin if and only if ∆p(A) ≤ 1 for each p ∈ X].
Proof. The theorem is proved in [10] (Theorems 4.1, 4.2, 4.3) for metric balleans, but the proof
can be easily adopted to the general case.
Given p ∈ X], we say [10] that a subset S ⊆ ¯̄p is ultrabounded with respect to p if there is α ∈ P
such that, for each q ∈ S and every Q ∈ q, we have B(Q,α) ∈ p.
We say that a subset A of X is
sparse if ∆p(A) is ultrabounded for each p ∈ X];
scattered if, for every Y ⊆ A, there is p ∈ Y ] such that ∆p(Y ) is ultrabounded.
To prove Theorems 5.2 and 5.3, one can adopt the arguments from [10] and [12].
Theorem 5.2. Let B = (X,P,B) be an unbounded ordinal ballean. For a subset A of X, the
following statements are equivalent:
(i) A is sparse;
(ii) for every unbounded subset Y of A, there exists β ∈ P such that, for every α ∈ P, we have
{y ∈ Y : BA(y, α) \BA(y, β) = ∅} 6= ∅.
Theorem 5.3. Let B = (X,P,B) be an unbounded ordinal ballean. For a subset A of X, the
following statements are equivalent:
(i) A is scattered;
(ii) for every unbounded subset Y of A, there exists β ∈ P such that, for every α ∈ P, we have
{y ∈ Y : BY (y, α) \BY (y, β) = ∅} 6= ∅.
A ballean B = (X,P,B) is called uniformly locally finite if, for every α ∈ P, there exists a
natural number n(α) such that |B(x, α)| ≤ nα for every x ∈ X. By [8] (Theorem 6), for every
locally finite ballean B = (X,P,B), there exists a group G of permutations of X such that B is
asymorphic to the ballean B(G,X,FG) (see Example 2.4), where FG is the ideal of finite subsets of
G.
The following statement is a part of Theorem 5.4 from [13].
Theorem 5.4. Let B = (X,P,B) be a uniformly locally finite ballean with the support X. A
subset A of X is scattered if and only if ∆p(A) is discrete for each p ∈ X].
Now we discuss a possibility generalization of Theorem 5.4 to arbitrary balleans.
Let B = (X,P,B) be a ballean. For p ∈ X] and α ∈ P, we set
B(p, α) = {q ∈ X] : B(P ′, α) ∈ q for each P ′ ∈ p}
and note that ¯̄p =
⋃
α∈P
B(p, α) and each subset B(p, α) is closed in ¯̄p.
We say that a point p ∈ X] is ball isolated if there exists P ′ ∈ p and α ∈ P such that if q ∈ ¯̄p
and P ′ ∈ q then q ∈ B(p, α). Applying Proposition 3.4, it is easy to verify that if p is ball isolated
then each point q ∈ ¯̄p is ball isolated. If B is uniformly locally finite then p is ball isolated if and
only if p is an isolated point of the subset ¯̄p of X].
Theorem 5.5. Let B = (X,P,B) be a ballean, A be a subset of X. If each point p ∈ A] is ball
isolated, then A is scattered.
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ULTRAFILTERS ON BALLEANS 1705
Proof. We say that a subset F ⊆ X] is invariant if p ∈ F and q‖p imply q ∈ F.
We take an arbitrary unbounded subset Y of A, denote by F the family of all closed invariant
subsets of X] and put
FY = {F ∈ Y ] : F ∈ F}.
By the Zorn lemma, there is a minimal by inclusion element M ∈ FY . We take an arbitrary
p ∈ M and show that ∆p(Y ) is ultrabounded. Assume the contrary and choose q ∈ cl ¯̄p such that
q /∈ B(p, α′) for each α′ ∈ P. Since q ∈M, by the minimality of M, p ∈ cl(¯̄q). By the assumption,
p is ball isolated. We choose corresponding P ′ ∈ p and α ∈ P. We pick q′ ∈ ¯̄q such that P ′ ∈ q′.
Since q′‖q, there is β ∈ P such that q ∈ B(q′, β) so B(P ′, β) ∈ q.
If p′ ∈ B(P ′, β) then, by Proposition 3.4, there is p′′ ∈ P ′ such that p′′ ∈ B(p′, β). We choose
γ ∈ P such that B(B(x, α), β) ⊆ B(x, γ) for each x ∈ X. Then q ∈ clB(p, γ) and q ∈ B(p, γ),
contradicting the choice of q.
Theorem 5.5 is proved.
Question 5.1. Let A be a scattered subset of X. Is every point p ∈ A] ball isolated? By
Theorem 5.4, this is so for each uniformly locally finite balleans but the question is open even for
metric balleans.
Recall that a topological space X is scattered if each nonempty subset Y of X has an isolated
point in Y.
Question 5.2. Let B be a ballean with the support X, A ⊆ X. Assume that each subspace
∆p(A), p ∈ A] is sparse in X]. Is A a sparse subset of B? By Theorem 5.4 and [8] (Theorem 6),
this is so for every uniformly locally finite ballean because in this case each ∆p(A) is discrete.
Given a ballean B = (X,P,B) and a subset A of X, we remind that a subset p̌ ∩A] is a corona
p-companion of A and characterize a size of A in terms of corona companions.
Theorem 5.6. Let B = (X,P,B) be an unbounded ordinal ballean, A ⊆ X. Then the following
statements hold:
(i) A is large if and only if p̌ ∩A] 6= ∅ for each p ∈ X];
(ii) A is thick if and only if there exists p ∈ X] such that p̌ ⊆ A];
(iii) A is thin if and only if |p̌ ∩A]| ≤ 1 for each p ∈ X].
Proof. (ii) Assume that A is thick. We may suppose that P is an infinite regular cardinal κ.
We choose a κ-sequence {yα : α < κ} in A such that B(yα, α) ⊆ A and B(yα, α) ∩ B(yβ, β) = ∅
for each α < β < κ. Then we pick an arbitrary ultrafilter p ∈ X] such that {yα : α < κ} ∈ p. By
Proposition 3.2, we have p̌ ∈ A].
Suppose that p̌ ⊆ A] for some p ∈ X]. Given any α < κ, there is P ∈ p such that B(P, α) ∈ A,
because otherwise, by Proposition 3.4, we can find q ∈ X] such that p||q and X \A ∈ q so ¯̄p * A].
Hence A is thick.
(i) It suffices to observe that A is large if and only if X \A is not thick and apply (ii).
(iii) Assume that there are two distinct ultrafilters p, q ∈ X] such that p ∼ q and A ∈ p, A ∈ q.
We choose P ∈ p, Q ∈ q such that P ⊆ A, Q ⊆ A and P ∩ Q = ∅. By Proposition 3.2, there is
α < κ such that B(P, α) ∩Q is unbounded. It follows that A is not thin.
If A is not thin, one can choose γ < κ and two κ-sequences {xα : α < κ} and {yα : α < κ} such
that xα 6= yα, yα ∈ B(xα, γ) and B(xα, γ)∩B(xβ, γ) = ∅ for all α < β < κ. We take an ultrafilter
p ∈ X] such that {xα : α < κ} ∈ p and use Proposition 3.4 to find q ∈ X] such that q||p and {yα :
α < κ} ∈ q. Then {p, q} ⊆ p̌ ∩A].
Theorem 5.6 is proved.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12
1706 I. V. PROTASOV, S. V. SLOBODIANIUK
Let G be an uncountable Abelian group, B = (G, [G]<ℵ0). By [9] (Proposition 4), the corona ǦB
is a singleton so Theorem 4.2 does not hold for B.
Question 5.3. Is Theorem 5.6 true for every unbounded normal ballean?
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Received 28.08.14
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12
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| id | umjimathkievua-article-2102 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| language | English |
| last_indexed | 2026-03-24T02:18:46Z |
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| publisher | Institute of Mathematics, NAS of Ukraine |
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| spelling | umjimathkievua-article-21022019-12-05T09:50:29Z Ultrafilters on balleans Ультрафiльтри на болеанах Protasov, I. V. Slobodianiuk, S. V. Протасов, І. В. Слободянюк, С. В. A ballean (equivalently, a coarse structure) is an asymptotic counterpart of a uniform space. We introduce three ultrafilter satellites of a ballean (namely, corona, companion, and corona companion), evaluate the basic cardinal invariants of the corona and characterize the subsets of balleans in terms of companions. Болеан (або груба структура) — це асимптотичний аналог рiвномiрного простору. За допомогою ультрафiльтрiв, визначено три супутники болеанiв (а саме, корону, компаньйон i короний компаньйон), знайдено оцiнки основних кардальних iнварiантiв корони та характеризовано пiдмножини болеанiв за допомогою компаньйонiв. Institute of Mathematics, NAS of Ukraine 2015-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2102 Ukrains’kyi Matematychnyi Zhurnal; Vol. 67 No. 12 (2015); 1698-1706 Український математичний журнал; Том 67 № 12 (2015); 1698-1706 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2102/1210 Copyright (c) 2015 Protasov I. V.; Slobodianiuk S. V. |
| spellingShingle | Protasov, I. V. Slobodianiuk, S. V. Протасов, І. В. Слободянюк, С. В. Ultrafilters on balleans |
| title | Ultrafilters on balleans |
| title_alt | Ультрафiльтри на болеанах |
| title_full | Ultrafilters on balleans |
| title_fullStr | Ultrafilters on balleans |
| title_full_unstemmed | Ultrafilters on balleans |
| title_short | Ultrafilters on balleans |
| title_sort | ultrafilters on balleans |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2102 |
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