Constructing balleans
A ballean is a set endowed with a coarse structure.We introduce and explore three constructions of balleans from a pregiven family of balleans: bornological products, bouquets, and combs. We analyze also the smallest and largest coarse structures on a set X compatible with a given bornology on X.
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Інститут прикладної математики і механіки НАН України
2018
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| Cite this: | Constructing balleans / T. Banakh, I. Protasov // Український математичний вісник. — 2018. — Т. 15, № 3. — С. 321-331. — Бібліогр.: 11 назв. — англ. |
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| citation_txt | Constructing balleans / T. Banakh, I. Protasov // Український математичний вісник. — 2018. — Т. 15, № 3. — С. 321-331. — Бібліогр.: 11 назв. — англ. |
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| description | A ballean is a set endowed with a coarse structure.We introduce and explore three constructions of balleans from a pregiven family of balleans: bornological products, bouquets, and combs. We analyze also the smallest and largest coarse structures on a set X compatible with a given bornology on X.
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Український математичний вiсник
Том 15 (2018), № 3, 321 – 331
Constructing balleans
Taras Banakh, Igor Protasov
Abstract. A ballean is a set endowed with a coarse structure. We
introduce and explore three constructions of balleans from a pregiven
family of balleans: bornological products, bouquets and combs. Also
we analyze the smallest and the largest coarse structures on a set X
compatible with a given bornology on X.
2010 MSC. 54E35.
Key words and phrases. Ballean, coarse structure, bornological prod-
uct, bouquet, comb.
1. Introduction
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−1 ∈ E , where E ◦E′ = {(x, y) :
∃z ((x, z) ∈ E, (z, y) ∈ E′)}, E−1 = {(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 =
∪
x∈X{x}×E[x], the
entourage E is uniquely determined by the family of balls {E[x] : x ∈ X}.
A subfamily B ⊂ E is called a base of the coarse structure E if each set
E ∈ E is contained in some B ∈ B.
The pair (X, E) is called a coarse space [11] or a ballean [8,10]. In [8]
every base of a coarse structure, defined in terms of balls, is called a ball
structure. We prefer the name balleans not only by the authors rights but
also because a coarse spaces sounds like some special type of topological
Received 18.09.2018
ISSN 1810 – 3200. c⃝ Iнститут прикладної математики i механiки НАН України
322 Constructing balleans
spaces. In fact, balleans can be considered as non-topological antipodes
of uniform topological spaces. Our compromise with [11] is in usage the
name coarse structure in place of the ball structure.
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 BX of all bounded subsets of X is a bornology on X. We recall
that a family B of subsets of a set X is a bornology if B contains the
family [X]<ω of all finite subsets of X and B is closed under finite unions
and taking subsets. A bornology B on a set X is called unbounded if
X /∈ 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 ] =
∪
y∈Y E[y].
Let (X, E), (X ′, E ′) be balleans. A mapping f : X → X ′ is called
coarse (or macrouniform) if for every E ∈ E there exists E′ ∈ E such
that f(E(x)) ⊆ E′(f(x)) for each x ∈ X. If f is a bijection such
that f and f−1 are coarse, then f is called an asymorphism. If (X, E)
and (X ′, E ′) contains large asymorphic subballeans, then they are called
coarsely equivalent.
For coarse spaces (Xα, Eα), α ∈ κ, their product is the Cartesian
product X =
∏
α∈αXα endowed with the coarse structure generated by
the base consisting of the entourages{(
(xα)α∈κ, (yα)α∈κ
)
∈ X ×X : ∀α ∈ κ (xα, yα) ∈ Eα
}
,
where (Eα)α∈κ ∈
∏
α∈κ Eα.
A class M of balleans is called a variety if M is closed under formation
of subballeans, coarse images and Cartesian products. For characteriza-
tion of all varieties of balleans, see [7].
Given a family F of subsets of X×X, we denote by E the intersection
of all coarse structures, containing each F ∪△X , F ∈ F, and say that E
is generated by F. It is easy to see that E has a base of subsets of the
form E1 ◦ E1 ◦ . . . ◦ En, where
E1, . . . , En ∈ {F ∪ F−1 ∪ {(x, y)} ∪ △X : F ∈ F, x, y ∈ X}.
By a pointed ballean we shall understand a ballean (X, E) with a
distinguished point e∗ ∈ X.
T. Banakh, I. Protasov 323
2. Metrizability and normality
Every metric d on a set X defines the coarse structure Ed 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 such that E = Ed.
Theorem 1 ([5]). 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 a subset 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 neighbourhoods).
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 is normal. In particular, every metrizable
ballean is normal, see [6].
A function f : X → R is called slowly oscillating if for any E ∈ E and
ε > 0, there exists a bounded subset B of X such that diam f(E[x]) < ε
for each x ∈ X \B.
Theorem 2 ([6]). A ballean (X, E) is normal if and only if for any two
disjoint asymptotically disjoint subsets Y,Z of X there exists a slowly
oscillating function f : X → [0, 1] such that f(Y ) ⊂ {0} and f(Z) ⊂ {1}.
For any unbounded bornology B on a set X the cardinals
add(B) = min{A ⊂ B :
∪
A /∈ B},
cov(B) = min{|C| : C ⊂ B,
∪
C = X} and
cof(B) = min{C ⊂ B : ∀B ∈ B ∃C ∈ C B ⊂ C}
are called the additivity, the covering number and the cofinality of B,
respectively. It is well-known (and easy to see) that add(B) ≤ cov(B) ≤
cof(B).
The following theorem was proved in [10, 1.4].
Theorem 3. If the product X × Y of balleans X,Y is normal then
add(BX) = cof(BX) = cof(BY ) = add(BY ).
Theorem 4. Let X be the Cartesian product of a family F of metrizable
balleans. Then the following statements are equivalent:
324 Constructing balleans
1. X is metrizable;
2. X is normal;
3. All but finitely many balleans from F are bounded.
Proof. We need only to show (2) ⇒ (3). Assume the contrary. Then
there exists a family (Yn)n<ω of unbounded metrizable balleans such
that the Cartesian product Y =
∏
n∈ω Yn is normal. On the other hand,
add(BY ) ≤ add(BY0) = ℵ0 and a standard diagonal argument shows that
cof(BY ) > ℵ0, contradicting Theorem 3.
3. Bornological products
Let {(Xα, Eα) : α ∈ A} be an indexed family of pointed balleans and
let B be a bornology on the index set A. For each α ∈ A by eα we denote
the distinguished point of the ballean Xα.
The B-product of the family of pointed balleans {Xα : α ∈ A} is the
set
XB =
{
(xα)α∈A ∈
∏
α∈A
Xα : {α ∈ A : xα ̸= eα} ∈ B
}
,
endowed with the coarse structure EB, generated by the base consisting
of the entourages{(
(xα)α∈A, (yα)α∈A
)
∈ XB ×XB : ∀α ∈ B (xα, yα) ∈ Eα
}
where B ∈ B and (Eα)α∈B ∈
∏
α∈B Eα.
For the bornology B = PA consisting of all subsets of the index set
A, the B-product XB coincides with the Cartesian product
∏
α∈AXα of
the coarse spaces (Xα, Eα).
If each Xα is the doubleton {0, 1} with distnguished point eα = 0,
then the B-product is called the B-macrocube on A. If |A| = ω and
B = [A]<ω, then we get the well-known Cantor macrocube, whose coarse
characterization was given by Banakh and Zarichnyi in [2].
For relations between macrocubes and hyperballeans, see [3], [9].
Theorem 5. Let B be a bornology on a set and let XB be the B-product
of a family of unbounded metrizable pointed balleans. Then the following
statements are equivalent:
1. XB is metrizable;
2. XB is normal;
T. Banakh, I. Protasov 325
3. |A| = ω and B = [A]<ω.
Proof. To see that (2) ⇒ (3), repeat the proof of Theorem 4.
Theorem 6. Let B be a bornology on a set A and let XB be the B-
product of a family {Xα : α ∈ A} of bounded pointed balleans which
are not singletons. The coarse space XB is metrizable if and only if the
bornology B has a countable base.
Proof. Apply Theorem 1.
Let X be a macrocube on a set A and Y be a macrocube on a set B,
A ∩ B = ∅. Then X × Y is a macrocube on A ∪ B and, by Theorem 3,
X × Y needs not to be normal.
Question 1. How can one detect whether a given macrocube is normal?
Is a B-macrocube on an infinite set A normal provided that B ̸= PA is a
maximal unbounded bornology on A?
Let {Xn : n < ω} be a family of finite balleans, B = [ω]<ω. By [10],
the B-product of the family {Xn : n < ω} is coarsely equivalent to the
Cantor macrocube.
Question 2. Let {Xα : α ∈ A} be a family of finite (bounded) pointed
balleans and let B be a bornology on A. How can one detect whether a
B-product of {Xα : α ∈ A} is coarsely equivalent to some macrocube?
4. Bouquets
Let B be a bornology on a set A and let {(Xα, Eα) : α ∈ A} be a
family of pointed balleans. The subballean∨
α∈A
Xα :=
{
(xα)α∈A ∈ XB : |{α ∈ A : xα ̸= eα}| ≤ 1
}
of the B-product XB is called the B-bouquet of the family {(Xα, Eα) :
α ∈ A}. The point e = (eα)α∈A is the distinguished point of the ballean∨
α∈AXα.
For every α ∈ A we identify the ballean Xα with the subballean
{(xβ)β∈A ∈ XB : ∀β ∈ A \ {α} xβ = eβ} of
∨
α∈AXα. Under such
identification
∨
α∈AXα =
∪
α∈AXβ and Xα ∩Xβ = {e} = {eα} = {eβ}
for any distinct indices α, β ∈ A.
Applying Theorem 1, we can prove the following two theorems.
326 Constructing balleans
Theorem 7. Let B be a bornology on a set A and let {Xα : α ∈ A}
be a family of unbounded pointed metrizable balleans. The B-bouquet∨
α∈AXα is metrizable if and only if |A| = ω and B = |A|<ω.
Theorem 8. Let B be a bornology on a set A and let {Xα : α ∈ A}
be a family of bounded pointed balleans, which are not singletons. The
B-bouquet
∨
α∈AXα is metrizable if and only if the bornology B has a
countable base.
Theorem 9. A bornological bouquet of any family of pointed normal
balleans is normal.
Proof. Let B be a bornology on a non-empty set A and X be the B-
bouquet of pointed normal balleans Xα, α ∈ A. Given two disjoint
asymptotically disjoint sets Y, Z ⊂ X, we shall construct a slowly oscil-
lating function f : X → [0, 1] such that f(Y ) ⊂ {0} and f(Z) ⊂ {1}.
The definition of the coarse structure on the B-bouquet ensures that
for every α ∈ A the subsets Y ∩ Xα and Z ∩ Xα are asymptotically
disjoint in the coarse space Xα, which is identified with the subspace
{(xβ) ∈ X : ∀β ∈ A\{α} xβ = eβ} of the B-bouquet X. By the normal-
ity of Xα, there exists a slowly oscillating function fα : Xα → [0, 1] such
that fα(Y ∩Xα) ⊂ {0} and fα(Z ∩Xα) ⊂ {1}. Changing the value of fα
in the distinguished point eα of Xα, we can assume that fα(eα) = fβ(eβ)
for any α, β ∈ A. Then the function f : X → [0, 1], defined by f�Xα = fα
for α ∈ A is slowly ascillating and has the desired property: f(Y ) ⊂ {0}
and f(Z) ⊂ {1}. By Theorem 2, the ballean X is normal.
5. Combs
Let (X, E) be a ballean and A be a subset of X. Let {(Xα, Eα) : α ∈
A} be a family of pointed balleans with the marked points eα ∈ Xα for
α ∈ A.
The bornology BX of the ballean (X, E) induces a bornology B :=
{B ∈ BX : B ⊂ A} on the set A. Let
∨
α∈AXα be the B-bouquet of the
family of pointed balleans {(Xα, Eα) : α ∈ A}, and let e we denote the
distinguished point of the bouquet
∨
α∈AXα.
For for every α ∈ A we identify the ballean Xα with the subballean
{(xβ)β∈A ∈
∨
α∈AXα : ∀β ∈ A \ {α} xβ = eβ} of
∨
α∈AXα. Then∨
α∈AXα =
∪
α∈AXα and Xα∩Xβ = {e} = {eα} = {eβ} for any distinct
indices α, β ∈ A.
The suballean
X ⊥⊥
α∈A
Xα := (X × {e}) ∪
∪
α∈A
({α} ×Xα)
T. Banakh, I. Protasov 327
of the ballean X ×
∨
α∈A
Xα is called the comb with handle X and spines
Xα, α ∈ A ⊂ X. We shall identify the handle X and the spines Xα
with the subsets X × {e} and {α} ×Xα in the comb X ⊥⊥
α∈A
Xα. It can
be shown that the comb X ⊥⊥
α∈A
Xα carries the smallest coarse structure
such that the identity inclusions of the balleans X and Xα, α ∈ A, into
X ⊥⊥
α∈A
Xα are macrouniform.
Theorem 10. The comb X ⊥⊥
α∈A
Xα is metrizable if the balleans X and
Xα, α ∈ A, are metrizable, and for each bounded set B ⊂ X the inter-
section A ∩B is finite.
Proof. Applying Theorem 7, we conclude that the bouquet
∨
α∈AXα is
metrizable. Then the comb X ⊥⊥
α∈A
Xα is metrizable being a subspace of
the metrizable ballean X ×
∨
α∈AXα.
By analogy with Theorem 9 we can prove
Theorem 11. The comb X ⊥⊥
α∈A
Xα is normal if the balleans X and Xα,
α ∈ A, are normal.
6. Coarse structures, determined by bornologies
Let B be a bornology on a set X. We say that a coarse structure E
on X is compatible with B if B coincides with the bornology BX of all
bounded subsets of (X, E).
The family of all coarse structures, compatible with a given bornology
B has the smallest and largest elements ⇓B and ⇑B.
The smallest coarse structure ⇓B is generated by the base consisting
of the entourages (B ×B) ∪△X , where B ∈ B.
The largest coarse structure ⇑B consists of all entourages E ⊆ X×X
such that E−1[B] ∪ E[B] ∈ B for every B ∈ B.
An unbounded ballean (X, E) is called
• discrete if E = ⇓BX ,
• ultradiscrete if X is discrete and its bornology BX is maximal by
inclusion in the family of all unbounded bornologies on X;
• maximal if its coarse structure is maximal by inclusion in the family
of all unbounded coarse structures on X;
• relatively maximal if E = ⇑BX .
328 Constructing balleans
It can be shown that an unbounded ballean (X, E) is discrete if and
only if for every E ∈ E there exists a bounded set B ⊂ X such that
E[x] = {x} for each x ∈ X\B. In [10, Chapter 3] discrete balleans are
called pseudodiscrete.
It is clear that each maximal ballean is relatively maximal. For maxi-
mal balleans, see [10, Chapter 10]. For any regular cardinal κ the ballean
(κ,⇑[κ]<κ) is maximal.
Each ultradiscrete ballean is both discrete and relatively maximal.
A ballean (X, E) is called ultranormal if X contains no two unboun-
ded asymptotically disjoint subsets. By [10, Theorem 10.2.1], every un-
bounded subset of a maximal ballean is large, which implies that each
maximal ballean is ultranormal. A discrete ballean is ultranormal if and
only if it is ultradiscrete.
Example 1. For every infinite set X, there exists a bornology B on X
such that ⇓B = ⇑B but the ballean (X,⇓B) = (X,⇑B) is not ultradiscrete.
Consequently, the ballean (X,⇓B) = (X,⇑B) is discrete and relatively
maximal but not ultranormal.
Proof. By Theorem 3.1.6 [4], there are two free ultrafilters p, q on X such
that for every function f : X → X and any P ∈ p and Q ∈ q we have
f(P ) /∈ q and f(Q) /∈ p. We put B = {B ⊆ X : B /∈ p, B /∈ q} and note
that B is a bornology on X.
To show that ⇓B = ⇑B, we need to check that for any entourage
E ∈ ⇑B, the set Y = {x ∈ X : E[x] ̸= {x}} belongs to the bornology B.
To derive a contradiction, assume that Y /∈ B. For every x ∈ Y choose
a point f(x) ∈ E[x] \ {x}. By Zorn’s Lemma, there exists a maximal
subset Z ⊂ Y such that Z ∩ f(Z) = ∅. By the maximality of Z, for any
y ∈ Y \ Z we get f(y) ∈ Z and hence f(Y \ Z) ⊂ Z. It follows from
Y /∈ B that Z /∈ B or Y \ Z /∈ B.
First assume that Z /∈ B. Then Z ∈ p or Z ∈ q. Without loss of
generality, Z ∈ p. Then f(Z) /∈ p and f(Z) /∈ q (by the choice of p, q).
Consequently, f(Z) ∈ B and Z ⊂ E−1[f(Z)] ∈ B, which is a desired
contradiction.
The case Y \ Z /∈ B can be considered by analogy.
Since X can be written as the union X = P ∪ Q of two disjoint
unbounded sets P ∈ p, Q ∈ q, the ballean (X,⇑B) is not ultradiscrete
and not ultranormal.
By a bornological space we understand a pair (X,BX) consisting of
a set X and a bornology BX on X. A bornological space (X,BX) is
unbounded if X /∈ BX . For two bornological spaces (X,BX) and (Y,BY )
T. Banakh, I. Protasov 329
their product is the bornological space (X × Y,B) endowed with the
bornology
BX×Y = {B ⊂ X × Y : B ⊂ BX ×BY for some BX ∈ BX , BY ∈ BY }.
The following theorem allows us to construct many examples of bor-
nological spaces (X,B) for which the coarse space (X,⇑B) is not normal.
Theorem 12. Let (X ×Y,B) be the product of two unbounded bornolog-
ical spaces (X,BX) and (Y,BY ). If cov(BY ) < add(BX), then the coarse
space (X × Y,⇑B) is not normal.
Proof. Fix any point (x0, y0) ∈ X × Y . Assuming that cov(BY ) <
add(BX), we shall prove that for a coarse structure E on X × Y is not
normal if E has the following three properties:
1. E is compatible with the bornology B;
2. for any BY ∈ BY there exists E ∈ E such that
X ×BY ⊂ E[X × {y0}];
3. for any BX ∈ BX there exists E ∈ E such that
BX × Y ⊂ E[{x0} × Y ].
It is easy to see that the coarse structure ⇑B has these three properties.
By the definition of the cardinal κ = cov(BY ), there there is a family
{Yα}α∈κ ⊂ BY such that
∪
α∈κ Yα = Y .
Assume that E is a coarse structure on X × Y satisfying the condi-
tions (1)–(3). First we check that the sets X × {y0} and {x0} × Y are
asymptotically disjoint in (X × Y, E). Given any entourage E ∈ E , we
should prove that the intersection E[X×{y0}]∩E[{x0}×Y ] is bounded.
By the condition (1), for every α ∈ κ the bounded set E−1[E[{x0}×Yα]]
is contained in the product Bα × Y for some bounded set Bα ∈ BX .
Since κ < add(BX), the union B<κ :=
∪
α∈κBα belongs to the bornol-
ogy BX . Given any point (u, v) ∈ E[X × {y0}] ∩ E[{x0} × Y ], find
x ∈ X and y ∈ Y such that (u, v) ∈ E[(x, y0)] ∩ E[(x0, y)]. Since
Y =
∪
α∈κ Yα, there exists α ∈ κ such that y ∈ Yα. Then (x, y0) ∈
E−1[E[(x0, y)]] ⊂ E−1[E[{x0} × Yα]] ⊂ Bα × Y ⊂ B<κ × Y and hence
(u, v) ∈ E[(x, y0)] ⊂ E[B<κ × {y0}], which implies that the intersection
E[X × {y0}] ∩ E[{x0} × Y ] ⊂ E[B<κ × {y0}]
is bounded in (X × Y, E).
Assuming that the coarse space (X×Y, E) is normal, we can find dis-
joint asymptotical neighborhoods U and V of the asymptotically disjoint
330 Constructing balleans
sets X ×{y0} and Y ×{x0}. By the condition (2), for every α ∈ κ there
exists an entourage Eα ∈ E such that X × Yα ⊂ Eα[X × {y0}]. Since U
is an asymptotic neighborhood of the set X ×{y0} in (X ×Y, E), the set
(X × Yα) \ U ⊂ Eα[X × {y0}] \ U is bounded in (X × Y, E). Now the
condition (1) implies that (X × Yα) \U ⊂ Dα × Y for some bounded set
Dα ∈ BX .
We claim that the family {Dα}α∈κ is cofinal in BX . Indeed, given any
bounded set D ∈ BX , use the condition (3) and find a entourage E ∈ E
such that D×Y ⊂ E[{x0}×Y ]. Since V is an asymptotic neighborhood
of the set {x0}×Y , the set E[{x0}×Y ]\V is bounded in (X×Y, E) and
the condition (1) ensures that it has bounded projection onto Y . Since
Y /∈ BY , we can find a point y ∈ Y such that X × {y} is disjont with
E[{x0}×Y ]\V . Find α ∈ κ with y ∈ Yα. Then (X×{y})∩E[{x0}×Y ] ⊂
V and hence
D×{y} ⊂ (X×y})∩E[{x0}×Y ] ⊂ (X×Yα)∩V ⊂ (X×Yα)\U ⊂ Dα×Y,
which yields the desired inclusion D ⊂ Dα. Therefore,
cof(BX) ≤ |{Dα}α∈κ| ≤ κ = cov(BY ) < add(BX),
which contradicts the known inequality add(BX) ≤ cof(BX).
References
[1] T. Banakh, I. Protasov, The normality and bounded growth of balleans,
https://arxiv.org/abs/1810.07979.
[2] T. Banakh, I. Zarichnyi, Characterizing the Cantor bi-cube in asymptotic cate-
gories // Groups Geom. Dyn., 5 (2011), 691–720.
[3] D. Dikranjan, I. Protasov, K. Protasova, N. Zava, Balleans, hyperballeans and
ideals, Appl. Gen. Topology (to appear).
[4] J. van Mill, An introduction to βω, in: Handbook of Set-Theoretic Topology
(K. Kunen, J. Vaughan eds), North Holland, 1984.
[5] I. Protasov, Metrizable ball structures // Algebra Discrete Math., 1 (2002), 129–
141.
[6] I. Protasov, Normal ball structures // Math. Stud., 10 (2003), 3–16.
[7] I. Protasov, Varieties of coarse spaces, Axioms, 7 (2018), 32.
[8] I. Protasov, T. Banakh, Ball Structures and Colorings of Groups and Graphs //
Math. Stud. Monogr. Ser., Vol. 11, VNTL, Lviv, 2003.
[9] I. Protasov, K. Protasova, On hyperballeans of bounded geometry // Europ. J.
Math., 4 (2018), 1515–1520.
T. Banakh, I. Protasov 331
[10] I. Protasov, M. Zarichnyi, General Asymptology // Math. Stud. Monogr. Ser.,
Vol. 12, VNTL, Lviv, 2007, p. 219.
[11] J. Roe, Lectures on Coarse Geometry, AMS University Lecture Ser 31, Provi-
dence, R.I., 2003, p. 176.
Contact information
Taras Banakh Jan Kochanowski University in Kielce,
Poland
Ivan Franko National University of Lviv,
Lviv, Ukraine
E-Mail: t.o.banakh@gmail.com
Igor Protasov Faculty of Computer Science and
Cybernetics of Taras Shevchenko
National University of Kyiv,
Kyiv, Ukraine
E-Mail: i.v.protasov@gmail.com
|
| id | nasplib_isofts_kiev_ua-123456789-169407 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1810-3200 |
| language | Russian |
| last_indexed | 2025-12-07T18:20:36Z |
| publishDate | 2018 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Banakh, T. Protasov, I. 2020-06-12T17:36:12Z 2020-06-12T17:36:12Z 2018 Constructing balleans / T. Banakh, I. Protasov // Український математичний вісник. — 2018. — Т. 15, № 3. — С. 321-331. — Бібліогр.: 11 назв. — англ. 1810-3200 2010 MSC. 54E35 https://nasplib.isofts.kiev.ua/handle/123456789/169407 A ballean is a set endowed with a coarse structure.We introduce and explore three constructions of balleans from a pregiven family of balleans: bornological products, bouquets, and combs. We analyze also the smallest and largest coarse structures on a set X compatible with a given bornology on X. ru Інститут прикладної математики і механіки НАН України Український математичний вісник Constructing balleans Article published earlier |
| spellingShingle | Constructing balleans Banakh, T. Protasov, I. |
| title | Constructing balleans |
| title_full | Constructing balleans |
| title_fullStr | Constructing balleans |
| title_full_unstemmed | Constructing balleans |
| title_short | Constructing balleans |
| title_sort | constructing balleans |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/169407 |
| work_keys_str_mv | AT banakht constructingballeans AT protasovi constructingballeans |