Pseudodiscrete balleans
A ballean B is a set X endowed with some family of subsets of X which are called the balls. The properties of the balls are postulated in such a way that a ballean can be considered as an asymptotic counterpart of a uniform topological space. A ballean is called pseudodiscrete if "almost all...
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| Цитувати: | Pseudodiscrete balleans / O.I. Protasova // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 4. — С. 81–92. — Бібліогр.: 9 назв. — англ. |
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irk-123456789-1573952019-06-21T01:28:02Z Pseudodiscrete balleans Protasova, O.I. A ballean B is a set X endowed with some family of subsets of X which are called the balls. The properties of the balls are postulated in such a way that a ballean can be considered as an asymptotic counterpart of a uniform topological space. A ballean is called pseudodiscrete if "almost all" balls of every pregiven radius are singletons. We give a filter characterization of pseudodiscrete balleans and their classification up to quasi-asymorphisms. It is proved that a ballean is pseudodiscrete if and only if every real function defined on its support is slowly oscillating. We show that the class of irresolvable balleans are tightly connected with the class of pseudodiscrete balleans. 2006 Article Pseudodiscrete balleans / O.I. Protasova // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 4. — С. 81–92. — Бібліогр.: 9 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 03E05, 03E75, 06A11, 54A05, 54E15.. http://dspace.nbuv.gov.ua/handle/123456789/157395 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
A ballean B is a set X endowed with some family
of subsets of X which are called the balls. The properties of the
balls are postulated in such a way that a ballean can be considered
as an asymptotic counterpart of a uniform topological space. A ballean is called pseudodiscrete if "almost all" balls of every pregiven
radius are singletons. We give a filter characterization of pseudodiscrete balleans and their classification up to quasi-asymorphisms. It
is proved that a ballean is pseudodiscrete if and only if every real
function defined on its support is slowly oscillating. We show that
the class of irresolvable balleans are tightly connected with the class
of pseudodiscrete balleans. |
| format |
Article |
| author |
Protasova, O.I. |
| spellingShingle |
Protasova, O.I. Pseudodiscrete balleans Algebra and Discrete Mathematics |
| author_facet |
Protasova, O.I. |
| author_sort |
Protasova, O.I. |
| title |
Pseudodiscrete balleans |
| title_short |
Pseudodiscrete balleans |
| title_full |
Pseudodiscrete balleans |
| title_fullStr |
Pseudodiscrete balleans |
| title_full_unstemmed |
Pseudodiscrete balleans |
| title_sort |
pseudodiscrete balleans |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2006 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/157395 |
| citation_txt |
Pseudodiscrete balleans / O.I. Protasova // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 4. — С. 81–92. — Бібліогр.: 9 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT protasovaoi pseudodiscreteballeans |
| first_indexed |
2025-07-14T09:49:54Z |
| last_indexed |
2025-07-14T09:49:54Z |
| _version_ |
1837615383454941184 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 4. (2006). pp. 81 – 92
c© Journal “Algebra and Discrete Mathematics”
Pseudodiscrete balleans
O. I. Protasova
Communicated by A. P. Petravchuk
Abstract. A ballean B is a set X endowed with some family
of subsets of X which are called the balls. The properties of the
balls are postulated in such a way that a ballean can be considered
as an asymptotic counterpart of a uniform topological space. A bal-
lean is called pseudodiscrete if "almost all" balls of every pregiven
radius are singletons. We give a filter characterization of pseudodis-
crete balleans and their classification up to quasi-asymorphisms. It
is proved that a ballean is pseudodiscrete if and only if every real
function defined on its support is slowly oscillating. We show that
the class of irresolvable balleans are tightly connected with the class
of pseudodiscrete balleans.
1. Ball structures and balleans
A ball structure is a triple B = (X,P,B), where X,P are nonempty sets
and, for any x ∈ X and α ∈ P , B(x, α) is a subset of X which is called a
ball of radius α around x. It is supposed that x ∈ B(x, α) for all x ∈ X,
α ∈ P . The set X is called the support of B, P is called the set of
radiuses.
Given any x ∈ X, A ⊆ X, α ∈ P , we put
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
2000 Mathematics Subject Classification: 03E05, 03E75, 06A11, 54A05,
54E15..
Key words and phrases: ballean, pseudodiscrete ballean, pseudobounded
ballean, slowly oscillating function, irresolvable ballean, asymorphism, quasi-
asymorphism.
82 Pseudodiscrete balleans
• lower symmetric if, for any α, β ∈ P , there exist α′, β′ ∈ P such
that, for every x ∈ X,
B∗(x, α′) ⊆ B(x, α), B(x, β′) ⊆ B∗(x, β);
• upper symmetric if, for any α, β ∈ P , there exist α′, β′ ∈ P such
that, for every x ∈ X,
B(x, α) ⊆ B∗(x, α′), B∗(x, β) ⊆ B(x, β′);
• lower multiplicative if, for any α, β ∈ P there exists γ ∈ P such
that, for every x ∈ X,
B(B(x, γ), γ) ⊆ B(x, α) ∩B(x, β);
• upper multiplicative if, for any α, β ∈ P there exists γ ∈ P such
that, for every x ∈ X,
B(B(x, α), β) ⊆ B(x, γ).
Let B = (X,P,B) be a lower symmetric, lower multiplicative ball
structure. Then the family
{
⋃
x∈X
B(x, α) ×B(x, α) : α ∈ P}
is a base of entourages for some (uniquely determined) uniformity on X.
On the other hand, if U ⊆ X × X is a uniformity on X, then the ball
structure (X,U , B) is lower symmetric and lower multiplicative, where
B(x, U) = {y ∈ X : (x, y) ∈ U}. Thus, the lower symmetric and lower
multiplicative ball structures can be identified with the uniform topolog-
ical spaces.
A ball structure is said to be a ballean if it is upper symmetric and
upper multiplicative. The balleans arouse independently in asymptotic
topology [1,4] under the name coarse structure and in combinatorics [5].
For good motivation to study the ballean related to metric space see the
survey [1].
Let B1 = (X1, P1, B1), B2 = (X2, P2, B2) be balleans. A mapping
f : X1 −→ X2 is called a ≺-mapping if, for every α ∈ P1, there exists
β ∈ P2 such that, for every x ∈ X1,
f
(
B1(x, α)
)
⊆ B2(f(x), β).
By the definition, ≺-mappings can be considered as the asymptotic
counterparts of the uniformly continuous mappings between the uniform
topological space.
O. I. Protasova 83
If f : X1 −→ X2 is a bijection such that f and f−1 are the ≺-
mappings, we say that the balleans B1 and B2 are asymorphic. IfX1 = X2
and the identity mapping id : X1 −→ X2 is a ≺-mapping, we write
B1 � B2. If B1 � B2 and B2 � B1, we write B1 = B2. Given an
arbitrary ballean B = (X,P,B), we can replace every ball B(x, α) to
B(x, α)
⋃
B∗(x, α) and get the same ballean on X,so in what follows we
assume that B(x, α) = B∗(x, α) for all x ∈ X, α ∈ P .
To determine the subject of this paper we need some more definitions.
Let B = (X,P,B) be a ballean.
A subset A of X is called bounded if there exists x ∈ X and α ∈ P
such that A ⊆ B(x, α). A ballean is called bounded if its support is
bounded.
Given any elements x, y ∈ X, we say that x, y are connected if there
exists α ∈ P such that y ∈ B(x, α). The connectedness is an equivalence
on X, so X disintegrates into connected components. A ballean B is
called connected if any two elements from X are connected.
A ballean B is called proper if B is connected and unbounded.
We use also the preodering ≤ on P defined by the rule: α ≤ β if and
only if B(x, α) ⊆ B(x, β) for every x ∈ X. A subset P ′ ⊆ P is called
cofinal if, for every α ∈ P , there exists β ∈ P ′ such that β ≥ α. The
minimal cardinality cf B of cofinal subsets is called cofinality of B.
Given a ballean B = (X,P,B), a subset Y ⊆ X is called large if there
exists α ∈ P such that X = B(Y, α). The large subsets of a ballean are
the asymptotic counterparts of the dense subsets of a uniform space.
2. Pseudodiscrete balleans
A ballean B = (X,P,B) is called discrete if B(x, α) = {x} for all x ∈
X, α ∈ P . Following [7], we say that B is pseudodiscrete if, for every
α ∈ P , there exists a bounded subset V of X such that B(x, α) = {x}
for every x ∈ X \V . Clearly, every discrete ballean is pseudodiscrete and
every bounded ballean is pseudodiscrete.
Let X be a set and let ϕ be a filter on X. Given any x ∈ X and
F ∈ ϕ, we put
Bϕ(x, F ) =
{
x, if x ∈ F ;
X \ F, if x ∈ X \ F ;
and denote by B(X,ϕ) the ballean (X,ϕ,Bϕ). Clearly, a subset V of X
is bounded in B(X,ϕ) if and only if either V is a singleton or X \V ∈ ϕ.
It follows that B(X,ϕ) is pseudodiscrete and B(X,ϕ) is bounded if and
only if X is a singleton. The list of connected components of B(X,ϕ) is
84 Pseudodiscrete balleans
X \
⋂
ϕ and {x}, x ∈
⋂
ϕ. Hence, B(X,ϕ) is connected if and only if
either
⋂
ϕ = ∅ or |X| = 1, B(X,ϕ) is proper if and only if
⋂
ϕ = ∅.
Every metric space (M,d) determines the metric ballean B(M,d) =
(M,R+, Bd), where R+ = {r ∈ R : r ≥ 0}, Bd(x, r) = {y ∈ M :
d(x, y) ≤ r}. A ballean B is called metrizable if B is asymorphic to some
metric ballean. By [6], B is metrizable if and only if B is connected and
cf B ≤ ℵ0. Hence, a ballean B(X,ϕ) is metrizable if and only if either
|X| = 1 of
⋂
ϕ = ∅ and ϕ has a countable base.
Theorem 1. Let B = (X,P,B) be an unbounded pseudodiscrete ballean.
Then there exists a filter ϕ on X such that B = B(X,ϕ).
Proof. First we show that at most one connected component of B is
not a singleton. Assume the contrary and choose two connected compo-
nents Y, Z such that |Y | > 1, |Z| > 1. Then we pick y, y′ ∈ Y, y 6= y′
and z, z′ ∈ Z, z 6= z′. Choose α ∈ P such that y′ ∈ B(y, α), z′ ∈ B(z, α).
Since B is pseudobounded, there exists a bounded subset V of X such
that |B(x, α)| = 1 for every x ∈ X \ V . Since every bounded subset
of an arbitrary ballean is contained in some connected component, then
either V
⋂
Y = ∅ or V
⋂
Z = ∅. If V
⋂
Y = ∅, then y ∈ X \ V and
|B(y, α)| > 1. If V
⋂
Z = ∅, then z ∈ X \ V and |B(z, α)| > 1. In both
cases we get a contradiction to the choice of V .
Let C be a union of all one-element connected components of X, A =
X \ C. If A is bounded, then B is determined by the filter ϕ = {Y ⊆
X : C ⊆ Y }. Suppose that the connected component A is unbounded.
Put ϕ = {X \ V : V is a bounded subset of A} and note that the union
of any finite family of bounded subsets of fixed connected component
is bounded, so ϕ is a filter on X. We show that B = B(X,ϕ). To
prove that B � B(X,ϕ), we take an arbitrary α ∈ P and choose a
bounded subset V such that |B(x, α)| = 1 for every x ∈ X \ V . If
V ⊆ A, we put U = B(V, α) and note that U is a bounded subset of
A, so X \ U ∈ ϕ and B(x, α) ⊆ Bϕ(x,X \ U). If V
⋂
A = ∅, then V
is contained in the connected component which is a singleton. Hence,
|B(x, α)| = 1 for all x ∈ X and B(x, α) ⊆ Bϕ(x, F ) for an arbitrary
F ∈ ϕ. To check that B(x, ϕ) � B, we fix an arbitrary F ∈ ϕ. By the
choice of ϕ, the subset V = X \ F is bounded in B. Choose α ∈ P
such that V ⊆ B(x, α) for every x ∈ V . Since Bϕ(x, F ) = {x} for every
x ∈ X \ V, Bϕ(x, F ) ⊆ B(x, α) for every x ∈ X. �
Let B = (X,P,B) be an arbitrary proper ballean. The family ϕ =
{X \ V : V is a bounded subsets of X} is a filter on X, so B has
the pseudodiscrete companion B(X,ϕ). By the definition, B(X,ϕ) is the
smallest (with respect to �) ballean on X such that every bounded subset
in B is bounded in B(X,ϕ).
O. I. Protasova 85
3. Subbaleans and factor-balleans
Let B = (X,P,B) be a ballean, Y be a nonempty subset of X. The
ballean BY = (Y, P,BY ), where BY (y, α)
⋂
X, is called a subballean of
X. Clearly, every subballean of pseudodiscrete ballean is pseudodiscrete.
A family F of subset of X is called uniformly bounded in B if there
exists α ∈ P such that F ⊆ B(x, α) for every x ∈ F . Let F be a
uniformly bounded partition of X. Given any F ∈ F and α ∈ P , we put
BF (F, α) = {F ′ ∈ F : F ′ ⊆ B(F, α)}. It is easy to check that the ball
structure B/F = (F , P,BF ) is a ballean which is called a factor-ballean
of B. We note also that B/F is a smallest (by �) ballean on F such that
the projection pr : X → F , where pr(x) = F if and only if x ∈ F , is a
≺-mapping.
Let X be a set, ϕ be a filter on X. A family F of subset of X
is uniformly bounded in the ballean B(X,ϕ) if and only if there exists
A ∈ ϕ such that, for every F ∈ F , either F ⊆ X\A or F is a singleton. In
view of Theorem 1, it follows that a factor-ballean of every pseudodiscrete
ballean is pseudodiscrete.
Let B1 = (X1, P1, B1), B2 = (X2, P2, B2) be balleans, f : X1 → X2
be a ≺-mapping. We consider the partition ker f of X1 determined by
the equivalence: x ∼ y if and only if f(x) = f(y). If the partition ker f is
uniformly bounded in B1, we get the canonical decomposition f = if ◦prf ,
where prf : X1 → ker f, if : ker f → X2. In this case, prf is a surjective
≺-mapping of B1 onto B1/ker f , if is a surjective ≺-mapping of B1/ker f
into B2.
4. Quasi-asymorphisms
Let B1 = (X1, P1, B1), B2 = (X2, P2, B2) be balleans. A mapping f :
X1 → X2 is called an asymorphic embedding of B1 into B2 if f is an
asymorphism between B1 and the subballean of B2 determined by the
subset f(X1) of X2.
A ≺-mapping f : X1 → X2 is called a quasi-asymorphic embedding
of B1 into B2 if, for every β ∈ P2, there exists α ∈ P1 such that, for all
x1, x2 ∈ X2, f(x1) ∈ B2(f(x2), β) implies x1 ∈ B1(x2, α). Equivalently,
f : X1 → X2 is a qiasi-asymorphic embedding if, for every uniformly
bounded family F1 of subsets of X1, the family f(F1) = {f(F ) : F ∈ F1}
is uniformly bounded in B2 and, for every uniformly bounded family F2
of subsets of X2, the family f−1(F2) = {f−1(F ) : F ∈ F2} is uniformly
bounded in B1. We note also that a quasi-asymorphic embedding f :
X1 → X2 is an asymorphic embadding if and only if f is injective. For
the case of metric ballean the notion of quasi-asymorphic embedding was
86 Pseudodiscrete balleans
introduced by Gromov [2] under the name uniform embedding.
Let f : X1 → X2 is a quasi-asymorphic embedding of B1 into B2.
Then the partition ker f is uniformly bounded in B1 and the mapping
if : ker f → X2 from the canonical decomposition f = if ◦ prf is an
asymorphic embedding of B1/ker f into B2. On the other hand, if some
factor-ballean of B1 admits an asymorphic embedding into B2, then B1
admits a quasi-asymorphic embedding into B2.
The next definition generalizes the notion of quasi-isometry between
metric spaces [3].
Let B1 = (X1, P1, B1), B2 = (X2, P2, B2) be balleans, f1 : X1 → X2
and f2 : X2 → X1 be ≺-mappings. We say that the pair (f1, f2) is a
quasi-asymorphism between B1 and B2 if there exist α ∈ P1, β ∈ P2 such
that, for all x ∈ X1, y ∈ X2,
f2f1(x) ∈ B(x, α), f1f2(y) ∈ B(y, β).
Automatically, in this case f1 and f2 are quasi-asymorphic embed-
dings and the subsets f1(X1) and f2(X2) are large in B2 and B1 respec-
tively.
Now we connect quasi-asymorphisms with quasi-asymorphic embed-
dings. Let f1 : X1 → X2 be a quasi-asymirphic embedding of B1 into B2
such that the subset f1(X1) of X2 is large in B2. We construct a mapping
f2 : X2 → X1 such that the pair (f1, f2) is a quasi-asymorphism between
B1 and B2. For every y ∈ f(X1), we choose some element g(y) ∈ f−1(y),
so we have the mapping g : f(X1) → X1. Since f(X1) is large in B2,
there exists β ∈ P2 such that B2(f(X1), β) = X2. To define the mapping
f2 : X2 → X1, we take an arbitrary z ∈ X2, choose y ∈ f(x1) such that
z ∈ B(y, α) and put f2(z) = g(y).
Clearly, every ballean, asymorphic to a pseudodiscrete ballean, is
pseudodiscrete, but a ballean, quasi-asymorphic to pseudodiscrete bal-
lean, needs not to be pseudodiscrete.
Theorem 2. For every ballean B = (X,P,B), the following statements
are equivalent:
(i) B is quasi-asymorphically embeddable to some pseudodiscrete bal-
lean;
(ii) B is quasi-asymorphic with some pseudodiscrete ballean;
(iii) there exists a uniformly bounded partition F of X such that the
factor-ballean B/F is pseudodiscrete.
Proof. (i)⇒(ii). If B is quasi-asymorphically embeddable to a pseu-
dodiscrete ballean B′, the B is quasi-asymorphic with some subballean of
B′ and it suffices to note that every subballean of pseudodiscrete ballean
is pseudodiscrete.
O. I. Protasova 87
(ii)⇒(iii). Let (f, g) be a quasi-asymorphism between B and some
pseudodiscrete ballean B′ with the support Y . We consider the canonical
decomposition f = if ◦ prf and note that if is an asymorphism between
B/ker f and the subballean of B′ defined by the subset f(X) of Y , so we
can put F = ker f .
(iii)⇒ (i). Since F is uniformly bounded, the projection pr : X → F
is a quasi-asymorphic embedding of B onto B/F . �
Using Theorem 2 we can easily construct, for every unbounded pseu-
dodiscrete ballean B, quasi-asymorphic ballean B′ which is not pseudodis-
crete. By Theorem 1, we may suppose that B = B(X,ϕ) where ϕ is a
filter on the support X of B. Let us take a disjoint family {Yx : x ∈ X}
of sets such that |Yx| > 1 for every x ∈ X. Put Y =
⋃
x∈X Yx and con-
sider the ballean B′ = (Y, ϕ,B′), where B′(y, F ) is defined by the rule: if
y ∈ Yx and x ∈ F , then B′(y, F ) = Yx, otherwise B′(y, F ) =
⋃
x∈X\F Yx.
Clearly, B′ is not pseudodiscrete, the family F = {Yx : x ∈ X} is
uniformly bounded in B′ and B′/F is asymorphic to B, so B′ is quasi-
asymorphic to B.
Let X,Y be sets, ϕ and ψ be filters on X and Y . We are going
to answer the question: when the pseudodiscrete balleans B(X,ϕ) and
B(Y, ψ) are quasi-asymorphic? We say that ϕ and ψ are equivalent if
there exist the subsets Φ ∈ ϕ, Ψ ∈ ψ and a bijection h : Φ → Ψ such
that, for a subset F ⊆ Φ, we have F ∈ ϕ if and only if h(F ) ∈ Ψ. If the
balleans B(X,ϕ) and B(Y, ψ) are asymorphic then ϕ and ψ are equivalent
with Φ = X, Ψ = Y .
Theorem 3. Let X,Y be sets, ϕ,ψ be filters on X and Y such that
⋂
ϕ =
⋂
ψ = ∅, then the balleans B(X,ϕ) and B(Y, ψ) are quasi-asymorphic if
and only if ϕ and ψ are equivalent.
Proof. Let B(X,ϕ) and B(Y, ψ) are quasi-asymorphic. We fix a quasi-
asymorphic embedding f : X → Y such that f(X) is large in Y . Since
the partition ker f of X is uniformly bounded in B(X,ϕ), there exists
Φ ∈ ϕ such that every element x ∈ Φ defines the element {x} of ker f .
It follows that the restriction f ′ of f to Φ is injective. Since X\Φ is
bounded in B(X,ϕ) and B(X,ϕ) is connected, Φ is large in B(X,ϕ),
hence f(Φ) is large in f(X). Since f(X) is large in B(Y, ψ), f(Φ) is large
in B(Y, ψ). It follows that f(Φ) ∈ ψ. Put Ψ = f(Φ). Then h : Φ →
Ψ is an asymorphism between the subballeans of B(X,ϕ) and B(X,ψ)
determined by the subsets Φ and Ψ. Hence, ϕ and ψ are equivalent.
Assume that ϕ and ψ are equivalent and Φ ∈ ϕ, Ψ ∈ ψ, h : Φ → Ψ
are corresponding sets and bijection. We take an arbitrary extension
f : X → Ψ and note that f is a quasi-asymorphic embedding of B(X,ϕ)
to B(Y, ψ). �
88 Pseudodiscrete balleans
Let B(X,ϕ) be a connected pseudodiscrete ballean. We take a symbol
∞ and put Ẋ = X
⋃
{∞}. Then we endow Ẋ with the topology τϕ in
the following way: every point x ∈ X is isolated in τϕ and the family
{F
⋃
{∞} : F ∈ ϕ} is a filter of neighborhoods of ∞ in τϕ. On the
other hand, let Y be a topological space with only one non-isolated point
y. Let ψ be a filter of neighborhoods of y. We put X = Y \{y}, ϕ =
{F\{y} : F ∈ ψ}. Then the ballean B(X,ϕ) is connected and (Ẋ, τϕ) is
homeomorphic to Y . Thus, we have defined the correspondence between
the class of connected pseudodiscrete balleans and the class of topological
spaces with only one non-isolated points.
Let X,Y be sets, ϕ,ψ be filters on X and Y such that
⋂
ϕ =
⋂
ψ = ∅.
Then the spaces (Ẋ, τϕ) and (Ẏ , τψ) are homeomorphic if and only if
B(X,ϕ) and B(Y, ψ) are asymorphic. By Theorem 3, B(X,ϕ) and B(Y, ψ)
are quasi-asymorphic if and only if the non-isolated points of (Ẋ, τϕ) and
(Ẏ , τψ) have homeomorphic neighborhoods.
5. Slowly oscillating functions
Let B = (X,P,B) be a ballean. A function f : X → R is called slowly
oscillating if , for α ∈ P and every ε > 0, there exists a bounded subset
V of X such that
diam f(B(x, α)) < ε
for every x ∈ X\V , where diamA = sup{|a− b| : a, b ∈ A}.
Theorem 4. For every ballean B = (X,P,B), the following statements
are equivalent:
(i) B is pseudodiscrete;
(ii) every function f : X → R is slowly oscillating;
(iii) every function f : X → {0, 1} is slowly oscillating.
Proof. (i)⇒(ii). Given α ∈ P and ε > 0, we take a bounded
subset V ⊆ X such that |B(x, α)| = 1 for every x ∈ X\V . Then
diamf(B(x, α)) = 0 for every x ∈ X\V , so diamf(B(x, α)) < ε.
The implication (ii)⇒(iii) is trivial.
(iii)⇒(i). First we show that at most one connected component of X
is not a singleton. Suppose not and choose two connected components
X0, X1 of such that |X0| > 1, |X1| > 1. Let x0, x
′
0 ∈ X0, x0 6= x′0 and
x1, x
′
1 ∈ X1, x1 6= x′1. Then we define a function f : X → {0, 1} by
the rule f(x0) = f(x1) = 0 and f(x) = 1 for every x ∈ X\{x0, x1}.
Pick α ∈ P such that x′0 ∈ B(x0, α), x′1 ∈ B(x1, α). If V is a bounded
subset of X, then there exists i ∈ {0, 1} such that V
⋂
Xi = ∅. Then
diamf(B(xi, α)) = 1, so f is not slowly oscillating. Hence, to prove that
B is pseudodiscrete, we may suppose that B is connected.
O. I. Protasova 89
Fix an arbitrary α ∈ P and choose a subset Y of X such that
{B(y, α) : y ∈ Y } is a maximal disjoint family. We put Y0 = {y ∈ Y :
|B(y, α)| = 1}, Y1 = Y \Y0. It suffices to show that X\Y0 is bounded.
Assume that Y1 is unbounded and consider the function f : X →
{0, 1}, defined by the rule: f |Y1
≡ 1, f |X\Y1
≡ 0. Given an arbi-
trary bounded subset V of X, we take y ∈ Y1 such that y /∈ V . Then
f(B(y, α)) = {0, 1}, so f is not slowly oscillating. Hence, Y1 is bounded.
Assume that X\Y0 is bounded and define a function f : X → {0, 1}
by the rule: f |Y0
≡ 1, f |X Y0
≡ 0. Pick β ∈ P such that B(B(x, α), α) ⊆
B(x, β) for every x ∈ X. By the assumption, f is slowly oscillating,
so there exists a bounded subset U of X such that diamf(B(x, β)) <
1
2 for every x ∈ X\U . Since B is connected and Y1, U are bounded,
the subset Y1
⋃
U is bounded. We put V = B(Y1
⋃
U, β). Since X\Y0
is unbounded and V is bounded, we can choose some z ∈ (X\Y0)\V .
Then diamf(B(z, β)) < 1
2 . Since z /∈ B(y1, β), by the choice of Y ,
there exists y ∈ Y0 such that B(z, α)
⋂
B(y, α) = ∅, so y ∈ B(z, β) and
diamf(B(z, β)) = 1, a contradiction. �
Under additional (but omited in formulation) assumption of connect-
edness of B the above theorem was proved in [7, Proposition 3.2].
Following [7], we say that a ballean B = (X,P,B) is pseudobounded if,
for every slowly oscillating function f : X → R, there exists a bounded
subset V of X such that f is bounded on X\V . To characterize the
pseudodiscrete pseudobounded balleans we use the Stone-Čech compact-
ifications.
Let X be a discrete space, βX be the Stone-Čech compactification of
X. We take the points of βX to be the ultrafilters on X with the points
of X identified with the principal ultrafilters. For every subset A ⊆ X,
we put Ā = {q ∈ βX : A ∈ q}. The topology of βX can be defined by
stating that the family {Ā : A ⊆ X} is a base for the open sets. For
every filter ϕ on X, the subset ϕ̄ =
⋂
{Ā : A ∈ ϕ} is closed in βX, and,
for every nonempty closed subset K ⊆ βG, there exists a filter ϕ on X
such that K = ϕ̄.
A filter ϕ on a set X is called countably complete if ϕ is closed under
countable intersections.
Theorem 5. Let X be a set and let ϕ be a filter on X. The ballean
B(X,ϕ) is pseudobounded if and only if the set ϕ̄ is finite and every
ultrafilter p ∈ ϕ̄ is countably complete.
Proof. Let B(X,ϕ) be pseudobounded, but ϕ̄ is infinite. Since the
space ϕ̄ is Hausdorf, we can choose a sequence (pn)n∈ω of elements of
ϕ̄ and a sequence (Pn)n∈ω of subsets of X such that, for every n ∈
ω, Pn ∈ pn and the family {Pn : n ∈ ω} is disjoint. We define a function
f : X → R by the rule: f |Pn
≡ n, n ∈ ω and f(x) = 0 for every
90 Pseudodiscrete balleans
x ∈ X\
⋃
n∈ω Pn. By Theorem 4, f is slowly oscillating. Let V be a
bounded subset of X. Then either X\V ∈ ϕ or V is singleton. In both
cases f |X V is unbounded, so B(X,ϕ) is not pseudodiscrete. Hence, ϕ̄ is
finite.
We show that every ultrafilter p ∈ ϕ̄ is countable complete. Other-
wise, we fix q ∈ ϕ̄ such that q is not countably complete and partition
X =
⋃
n∈ωXn so that Xn 6= q for every n ∈ ω. Define a function
f : X → R by the rule f |Xn
≡ n, n ∈ ω. Let V be an arbitrary bounded
subset of X. If there exists m ∈ ω such that Xn ⊆ V for every n ≥ ω,
then Xi ∈ q for some i < n, contradicting to the choice of {Xn : n ∈ ω}.
Hence, Xn\V = ∅ for infinitely many n ∈ ω and f |Xn\V is unbounded
and B is not pseudobounded.
Suppose that ϕ̄ is finite and every ultrafilter p ∈ ϕ̄ is countably com-
plete. We fix an arbitrary function f : X → R and put Xn = {x ∈ X :
n ≤ |f(x)| < n+ 1}, n ∈ ω. For every p ∈ ϕ̄, we pick n(p) ∈ ω such that
Xn(p) ∈ p and put Y =
⋃
p∈ϕ̄Xn(p). Then Y ∈ ϕ and f |Y is bounded, so
B(X,ϕ) is pseudobounded. �
In view of Theorem 5, there exists a proper pseudodiscrete pseu-
dobounded ballean on a set X if and only if there exists a countably
complete free ultrafilter on X, i.e. the cardinal |X| is Ulam-measurable.
It is well-known that the first Ulam-measurable cardinal is measurable.
Hence, the existence of proper pseudodiscrete pseudobounded ballean is
equivalent to the set-theoretical axiom MC.
6. Pseudodiscretness and resolvability
A ballean B = (X,P,B) is called irresolvable if X can not be partitioned
to two large subsets. For resolvability of balleans see [8]. Clearly, a
ballean B is irresolvable if and only if at least one connected component
of B is irresolvable, a bounded ballean is irresolvable if and only if it is a
singleton, so the irresolvability problem is reduced to the class of proper
balleans.
We show that irresolvability is tightly connected with pseudodiscret-
ness.
Theorem 6. For a proper ballean B = (X,P,B), the following state-
ments are equivalent:
(i) B is irresolvable;
(ii) for every α ∈ P , the subset Xα = {x ∈ X : |B(x, α)| = 1} is
unbounded;
(iii) there exists a filter ϕ on X such that
⋂
ϕ = ∅ and B � B(X,ϕ).
Proof. (i)⇒(ii). Assume the contrary: the subset Xα is bounded for
some α ∈ P . We take a subset Y of X such that |B(y, α)| > 1, y ∈ Y
O. I. Protasova 91
and the family {B(y, α) : y ∈ Y } is maximal disjoint. For every y ∈ Y ,
we pick y′ ∈ B(y, α), y′ 6= y and put Y ′ = {y′ : y ∈ Y }. Then Y, Y1 are
disjoint large subsets of X and we get a contradiction to irresolvability
of B.
(ii)⇒(iii). If α, α′ ∈ P and β ≥ α, β ≥ α′, then Xβ ⊆ Xα
⋂
Xα′ .
It follows that the family {Xα : α ∈ P} is a base for some filter ϕ on
X. Since B is connected,
⋂
ϕ = ∅. For any x ∈ X, α ∈ P , we have
B(x, α) ⊆ Bϕ(x,X Xα), so B � B(X,ϕ).
(iii)⇒(i). Let A be a subset of X. If A is large in B, then A is large
in B(X,ϕ). If A is large in B(X,ϕ), then A ∈ ϕ. It follows that any two
large in B subsets of X meets, so B is irresolvable. �
Let B = (X,P,B) be a ballean, Y ⊆ X. We say that subbalean BY of
B is almost isolated (almost invariant in terminology of [9]) if, for every
α ∈ P , the subset B(Y, α) Y is bounded in B.
Lemma 1. Let B = (X,P,B) be a proper ballean, Y ⊆ X and subballean
BY is unbounded and almost isolated in B. If BY is irresolvable, then B
is irresolvable.
Proof. Given an arbitrary α ∈ P , it suffices to fined y ∈ Y such
that B(y, α) ⊆ Y and |B(y, α)| = 1. Assume that it does not hold. Put
Y0 = {y ∈ Y : B(y, α) * Y }, Y1 = {y ∈ Y : B(y, α) ⊆ Y }. Since BY is
almost isolated in B, Y0 is bounded. By the assumption, |B(y, α)| > 1
for every y ∈ Y1. We choose a subset Z ⊆ Y1 such that the family
{B(z, α) : z ∈ Z} is maximal disjoint. For every z ∈ Z, we take an
arbitrary z′ ∈ B(z, α), z 6= z′. Put Z ′ = {z′ : z ∈ Z}. If x ∈ Y1, then
B(x, α)
⋂
B(z, α) 6= ∅ for some z ∈ Z. It follows that Z and Z ′ are
large in Y1. Since Y0 is bounded, Y = Y0
⋃
Y1 and BY is connected, then
Z and Z ′ are disjoint large subsets in BY , so we get a contradiction to
irresolvability of BY . �
Following [7], we say that a ballean B = (X,P,B) is ordinal if there
exist a cofinal well-ordered (with respect to ≤) subset of P . Note that
every metrizable ballean is ordinal.
Lemma 2. Let B = (X,P,B) be a proper irresolvable ordinal ballean.
Then there exists an unbounded subset Y of X such that the subballean
BY is pseudodiscrete and almost isolated in B.
Proof. We may suppose that P is well-ordered. Let |P | = γ. We
identify P with the set of ordinals {α : α < γ}. Replacing P by some
its cofinal subset, we may assume that γ is a regular cardinal. For every
α ∈ P , we put Yα = {y ∈ X : |B(y, α)| = 1}. By Theorem 6, Yα is
unbounded. Since γ is regular, every subset of X of cardinality < γ is
bounded. This observation allows us to construct an injective transfinite
92 Pseudodiscrete balleans
sequence (yα)α<γ of elements of X such that, for every α < γ, we have
yα ∈ Yα, B(yα, α)
⋂
(
⋃
λ<α
B(yλ, λ)) = ∅.
Put Y = {yα : α < γ}. Then the subset {yλ : λ < α} is bounded
in B and B(yβ , α) ⊆ Y, |B(yβ , α)| = 1 for every β ≥ 1. Hence, BY is
pseudodiscrete and almost isolated in B. �
We do not know whether Lemma 2 is true without the ordinalily
assumption on B.
Theorem 7. Let B = (X,P,B) be a proper ordinal ballean. Then B
is irresolvable if and only if there exists a subset Y of X such that the
subbalean BY is pseudodiscrete and almost invariant in B.
Proof. Apply Lemmas 1 and 2. �
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[4] Mitchener P.D. Coarse Homology Theories, Algebr. Geom. Topology, 1 (2001),
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[5] Protasov I., Banakh T. Ball Structures and Colorings of Graphs and Groups,
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129-141.
[7] Protasov I.V. Normal ball structures, Math. Stud., 20 (2003), 3 - 16.
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Contact information
O. I. Protasova Department of Cybernetics, Kyiv National
University, Volodimirska 64, Kiev 01033,
UKRAINE
E-Mail: polla@unicyb.kiev.ua
Received by the editors: 11.05.2003
and in final form 29.03.2007.
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