Balleans and G -spaces
We show that every ballean (equivalently, coarse structure) on a set $X$ can be determined by some group $G$ of permutations of $X$ and some group ideal $\mathcal{I}$ on $G$. We refine this characterization for some basic classes of balleans: metrizable, cellular, graph, locally finite, and uniform...
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| author | Petrenko, O. V. Protasov, I. V. Петренко, О. В. Протасов, І. В. |
| author_facet | Petrenko, O. V. Protasov, I. V. Петренко, О. В. Протасов, І. В. |
| author_sort | Petrenko, O. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:30:02Z |
| description | We show that every ballean (equivalently, coarse structure) on a set $X$ can be determined by some group $G$ of permutations of $X$ and some group ideal $\mathcal{I}$ on $G$.
We refine this characterization for some basic classes of balleans: metrizable, cellular, graph, locally finite, and uniformly locally finite. Then we show that a free
ultrafilter $\mathcal{U}$ on $\omega$ is a $T$-point with respect to the class of all metrizable locally finite balleans on $\omega$ if and only if $\mathcal{U}$ is a $Q$-point.
The paper is concluded with а list of open questions. |
| first_indexed | 2026-03-24T02:26:10Z |
| format | Article |
| fulltext |
UDC 512.5
O. V. Petrenko, I. V. Protasov (Kyiv. Nat. Taras Shevchenko Univ.)
BALLEANS AND G-SPACES
БОЛЕАНИ ТА G-ПРОСТОРИ
We show that every ballean (equivalently, coarse structure) on a set X can be determined by some group G of permutations
of X and some group ideal I on G. We refine this characterization for some basic classes of balleans: metrizable, cellular,
graph, locally finite, and uniformly locally finite. Then we show that a free ultrafilter U on ω is a T -point with respect to
the class of all metrizable locally finite balleans on ω if and only if U is a Q-point. The paper is concluded with а list of
open questions.
Доведено, що кожен болеан (еквiвалентно, груба структура) на множинi X може бути визначений деякою групою
пiдстановок G множини X та деяким груповим iдеалом I на G. Цю характеризацiю уточнено для деяких основних
класiв болеанiв: метризовних, стiльникових, графових, локально скiнченних, рiвномiрно локально скiнченних. Далi
ми доводимо, що вiльний ультрафiльтр U на ω є T -точкою вiдносно класу метризовних локально скiнченних
болеанiв на ω тодi i тiльки тодi, коли U є Q-точкою. Насамкiнець наведено список вiдкритих проблем.
Following [5, 6], we say that a ball structure is a triple B = (X,P,B), where X, P are non-empty
sets and, for every 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 and α ∈ P. The set X is called the support
of B, P is called the set of radii.
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 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).
A ballean B on X can also be determined in terms of entourages of the diagonal ∆X of X ×X,
in this case it is called a coarse structure [7] (Definition 2.3). Let E be a family of subsets of X ×X.
The pair (X, E) is a coarse structure if
∆X ⊂ E for each E ∈ E ;
if E ∈ E and ∆X ⊆ E′ ⊆ E then E′ ∈ E ;
if E1, E2 ∈ E then E1 ∪ E2 ∈ E ;
if E ∈ E then E−1 ∈ E where E−1 = {(y, x) : (x, y) ∈ E};
if E1, E2 ∈ E then E1 ◦ E2 ∈ E where E1 ◦ E2 = {(x, y) : (x, z) ∈ E1, (z, y) ∈ E2 for some
z ∈ X}.
Each ballean B = (X,P,B) defines a coarse structure (X, E) where the family E of entourages
is defined by the rule:
c© O. V. PETRENKO, I. V. PROTASOV, 2012
344 ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
BALLEANS AND G-SPACES 345
E ∈ E ⇔ ∃α ∈ P ∀x ∈ X ∀y ∈ X : y ∈ B(x, α)⇒ (x, y) ∈ E.
On the other hand, each coarse structure (X, E) determines the ballean (X, E , B),
where B(x,E) = {y ∈ X : (x, y) ∈ E}.
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). If there exists a bijection f : X1 → X2 such that f and f−1 are ≺-mappings, then
B1 and B2 are called asymorphic.
Let G be a group, I be an ideal in the Boolean algebra PG of all subsets of G, i.e., if A,B ∈ I
and A′ ⊆ A then A ∪B ∈ I and A′ ∈ I. An ideal I is called a group ideal if, for all A,B ∈ I, we
have AB ∈ I and A−1 ∈ I.
Now let X be a G-space with the action G ×X → X, (g, x) 7→ gx, and let I be a group ideal
on G. We define a ballean B(G,X, I) as triple (X, I, B) where B(x,A) = Ax∪ {x} for all x ∈ X,
A ∈ I.
In Section 1 we show that every ballean B with the support X is asymorphic to the ballean
B(G,X, I) for some group G of permutations of X and some group ideal I on G. Then we refine this
statement to some basic classes of balleans: metrizable, cellular, graph, locally finite and uniformly
locally finite.
Let B = (X,P,B) be a ballean. A subset F ⊆ X is called bounded if there exist x ∈ X and
α ∈ P such that F ⊆ B(x, α). A subset T ⊆ X is thin if, for every α ∈ P, there exists a bounded
subset F such that |B(x, α) ∩ T | 6 1 for each x ∈ X \ F.
Given a class K of balleans on ω = {0, 1, . . .}, a free ultrafilter U on ω is said to be a T -
point with respect to K if, for every ballean B ∈ K, there exists U ∈ U such that U is thin in B.
By Theorem 6, a T -point in ω∗ defined in [3] is exactly a T -point with respect to the class of all
metrizable uniformly locally finite balleans on ω. By [3] (Theorem 3), an ultrafilter U ∈ ω∗ is a
T -point with respect to the class of all metrizable balleans on ω if and only if U is selective.
In Section 2 we prove that an ultrafilter U ∈ ω∗ is a T -point with respect to the class of all
metrizable uniformly locally finite balleans on ω if and only if U is a Q-point. We give also some
“sequential” characterization of T -points with respect to the class of all metrizable uniformly locally
finite balleans on ω.
We conclude the paper with some comments and open questions in Section 3.
1. Balleans and G-spaces.
Theorem 1. Every ballean B with the support X is asymorphic to the ballean B(G,X, I) for
some subgroup G of the group SX of all permutations of X and some group ideal I of G.
Proof. Let E be a family of entourages of the diagonal ∆X of X × X which determines B.
For each pair (x, y) ∈ X × X, let π(x, y) denote the permutation of X swapping x, y and acting
identically on X \ {x, y}. Given any E ⊆ X ×X, we put
AE = {e, π(x, y) : (x, y) ∈ E},
where e is the identity permutation. We denote by G the subgroup of SX generated by ∪{AE : E ∈
∈ E}. To construct the ideal I, we put F0 = {AE : E ∈ E} and, for each n ∈ ω,
Fn+1 = Fn ∪ F−1
n ∪ {FF ′ : F, F ′ ∈ Fn}.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
346 O. V. PETRENKO, I. V. PROTASOV
Then ∪n∈ωFn is a base for I, i.e., I = {A ⊆ G : ∃F ∈ ∪n∈ωFn such that A ⊆ F}.
Given any E,E′ ∈ E , we have
(1) E = {(x, y) : x ∈ X, y ∈ AEx};
(2) E ◦ E′ = {(x, y) : x ∈ X, y ∈ AE◦E′x}.
By (1), the identity mapping id : X → X is a ≺-mapping from B to B(G,X, I). Using (2) and
inductive argument, we see that, for each A ∈ Fn, {(x, y) : x ∈ X, y ∈ Ax} ∈ E . Hence, id is a
≺-mapping from B(G,X, I) to B, so B and B(G,X, I) are asymorphic.
Theorem 1 is proved.
A ballean B = (X,P,B) is called connected if, for all x, y ∈ X, there is α ∈ P such that
y ∈ B(x, α). We observe that a ballean B(G,X, I) is connected if and only if, for all x, y ∈ X there
is A ∈ I such that x ∈ Ay. In particular, if B(G,X, I) is connected then G acts transitively on X.
Each metric space (X, d) determines the ballean B(X, d) = (X,R+, Bd) where R+ = {r ∈
∈ R : r > 0}, Bd(x, r) = {y ∈ XG : d(x, y) 6 r}. A ballean B is called metrizable if B is
asymorphic to B(X, d) for an appropriate metric space (X, d). By [6] (Theorem 2.1.1), a ballean B
is metrizable if and only if B is connected and its set of radii P has a countable subset cofinal in the
natural preordering of P.
Repeating arguments proving Theorem 1, we get the following theorem.
Theorem 2. Every metrizable ballean B with the support X is asymorphic to the ballean
B(G,X, I) for some subgroup G of SX and some group ideal I with countable base such that, for
all x, y ∈ X, there is A ∈ I such that y ∈ Ax.
A ballean B = (X, E) is called cellular [6, p. 42] if, for each entourage E′ ∈ E there is an
entourage E such that E′ ⊂ E, E = E−1 and E ◦ E = E. By [6] (Theorem 3.1.3), B is cellular if
and only if asdim B = 0.
Theorem 3. Every cellular ballean B with the support X is asymorphic to the ballean
B(G,X, I) for some subgroup G of SX and some group ideal I on G which has a base consisting
of subgroups.
Proof. To apply arguments proving Theorem 1, it suffices to note that if E ∈ E , E = E−1 and
E ◦ E = E then
{(x, y)G : x ∈ X, y ∈ AEx} = {(x, y)G : x ∈ X, y ∈ 〈AE〉x},
where 〈AE〉 is a subgroup of SX generated by AE .
Theorem 3 is proved.
Every connected graph Γ can be considered as a metric space (VΓ, dΓ) where VΓ is the set of
vertices of Γ, dΓ is the path metric on VΓ. A ballean B is called a graph ballean [6, p. 79] if B is
asymorphic to the ballean B(VΓ, dΓ) for an appropriate connected graph Γ. By [6] (Theorem 5.1.1),
a ballean B = (X, E) is a graph ballean if and only if B is connected and there exists E ∈ E such
that E = E−1 and, for every E′ ∈ E , there is n ∈ ω such that E′ = En where En is a product of n
copies of E.
Theorem 4. Every graph ballean B with the supportX is asymorphic to the ballean B(G,X, I)
for some subgroup G of SX and some group ideal I having a member A ∈ I such that {AnG : n ∈
∈ ω} is a base for I and, for all x, y ∈ X, there is n ∈ ω such that y ∈ Anx.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
BALLEANS AND G-SPACES 347
A ballean B = (X,P,B) is called locally finite if each ball B(x, α), x ∈ X, α ∈ P is finite. The
next theorem was suggested by Taras Banakh.
Theorem 5. Every locally finite ballean B with the support X is asymorphic to the ballean
B(G,X, I) for some subgroup G of SX and some group ideal I on G with a base consisting of
subsets compact in the topology of pointwise convergence on SX .
Proof. It suffices to observe that a subset F ⊂ SX is compact if and only if F is closed and
the orbit Fx of each point x ∈ X is finite, and the subset AE defined in the proof of Theorem 1 is
compact.
A ballean B = (X,P,B) is called uniformly locally finite if, for each α ∈ P, there exists n ∈ ω
such that |B(x, α)| 6 n for every x ∈ X.
Theorem 6. Every uniformly locally finite ballean B with the support X is asymorphic to the
ballean B(G,X,FG) for some subgroup G of SX , FG is the ideal of all finite subsets of G.
Proof. [4] (Theorem 1).
2. Around T -points. Let K be a class of balleans with the support ω = {0, 1, . . .}, ω∗ be the
space of all ultrafilters on ω. We say that an ultrafilter U ∈ ω∗ is a T -point with respect to K if, for
every ballean B = (ω, P,B) from K, U has a member U ∈ U which is thin in B, i.e., for every
α ∈ P, there exists a bounded subset V ⊂ ω such that |B(x, α) ∩ U | 6 1 for each x ∈ U \ V.
We recall that an ultrafilter U ∈ ω∗ is
selective if, for every partition P of ω, either some block of P is a member of U , or there is
U ∈ U such that |U ∩ P | 6 1 for each P ∈ P;
P -point if, for every partition P of ω, either some block of P is a member of U , or there is
U ∈ U such that U ∩ P is finite for each P ∈ P;
Q-point if, for every partition P of ω into finite subsets, there is U ∈ U such that |U ∩ P | 6 1
for each P ∈ P.
Theorem 7. An ultrafilter U ∈ ω∗ is a T -point with respect to the class of all metrizable
balleans on ω if and only if U is selective.
Proof. [3] (Theorem 3).
Theorem 8. An ultrafilter U ∈ ω∗ is a T -point with respect to the class of all metrizable
locally finite balleans on ω if and only if U is a Q-point.
Proof. Let U be a Q-point, d be a locally finite metric on ω. We fix x0 ∈ ω, and put
X0 = {x0}, Xn+1 = Bd(x0, (n+ 1)2) \Bd(x0, n
2), n ∈ ω,
Y0 =
⋃
n∈ω
X2n, Y1 =
⋃
n∈ω
X2n+1.
Since U is a Q-point and each subset Xn is finite, there exist U ∈ U and i ∈ {0, 1} such that U ⊆ Yi
and |U ∩Xn| 6 1 for each n ∈ ω. Then for each r > 0, the set {x ∈ UG : |Bα(x, r) ∩ U| > 1} is
finite, so U is thin in B(ω, d).
Now let U be a T -point in the class of all metrizable locally finite balleans on ω, {PnG : n ∈ ω}
be a partition of ω into finite subsets. We define a metric d on ω by the rule:
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
348 O. V. PETRENKO, I. V. PROTASOV
d(x, y) =
0, if x = y;
1, if x 6= y, x, y ∈ Pn;
max{n,m}, if x ∈ Pn, y ∈ Pm, n 6= m.
Since (ω, d) is locally finite, we can choose a subset U ∈ U which is thin in (ω, d). By the definition
of thin subset, there exists a bounded subset Y of (ω, d) such that Bd(x, 1) ∩ Bd(x′, 1) = ∅ for all
distinct x, x′ ∈ U \ Y. Since Y is finite, we conclude that U is a Q-point.
Theorem 8 is proved.
A situation with T -points with respect to the class of all metrizable uniformly locally finite
balleans on ω is much more delicate. It is proved in [3] that all P -points and all Q-points are T -
points, but it is unknown whether there exists a T -point in ZFC without additional set-theoretical
assumptions. Now we give some characterization of T -points based on sequences of coverings of ω.
Let us say that a covering F of ω is uniformly bounded if there exists a natural number n such
that, for each x ∈ ω,
|st(x,F)| 6 n,
where st(x,F) =
⋃
{F ∈ FG : x ∈ F}.
Theorem 9. An ultrafilter U ∈ ω∗ is a T -point with respect to the class K of all metrizable
uniformly locally finite balleans on ω if and only if, for every sequence (Fn)n∈ω of uniformly bounded
coverings of ω, there exists U ∈ U such that, for each n ∈ ω, |F ∩ U | 6 1 for all but finitely many
F ∈ Fn.
Proof. We assume that U is a T -point with respect to K and fix a sequence (Fn)n∈ω of uniformly
bounded coverings of ω. We consider a ball structure B = (ω, ω,B), where B(x, n) = st(x,Fn), and
take the ballean envelope env B = (ω, P,B′) of B, the smallest ballean on ω such that B ≺ env B
(see [6, p. 191]). By the description, env B is uniformly locally finite and P is countable. Joining
the elements 0, n to the first member of Fn, we may suppose that env B is connected. By [6]
(Theorem 2.1.1), env B is metrizable. Since U is a T -point with respect to K and env B ∈ K, there
is a subset U ∈ U which is thin in env B. We note that ω ⊆ P and B(x, n) = B′(x, n). Hence, for
each n ∈ ω, there exists a finite subset V of ω such that |U ∩ st(x,Fn)| 6 1 for each x ∈ U \ V. It
follows that |U ∩ F | 6 1 for all but finitely many F ∈ Fn.
To prove the converse statement, we take an arbitrary uniformly locally finite metric d on ω, and
put
Fn = (Bd(m,n))m∈ω, n ∈ ω.
Since each covering Fn is uniformly bounded, there is U ∈ U such that, for each n ∈ ω, |U ∩
∩Bd(m,n)| 6 1 for all but finitely many m ∈ ω. This means that U is thin in B(ω, d).
Theorem 9 is proved.
We conclude this section with the following observation suggested by Sergiy Slobodianiuk.
Theorem 10. An ultrafilter U on ω is a P -point if and only if, for every metric d on ω, either
some member of U is bounded in (X, d) or there is U ∈ U such that (U, d) is locally finite.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
BALLEANS AND G-SPACES 349
Proof. Assume that U is a P -point, fix x0 ∈ ω and write ω as a union ω =
⋃
n∈ω Bd(x0, n).
It follows from a definition of a P -point that there is U ∈ U such that U is contained in some ball
Bd(x0, n) or U meets each ball in finite number of points. Thus, U is bounded in the first case and
(U, d) is locally finite in the second case.
To prove the converse statement, we use the metric d defined by the partition {PnG : n ∈ ω} of
ω in Theorem 8. Let U be a subset of X. If U is bounded in (X, d) then U is contained in the union
of finitely many blocks. If (U, d) is locally finite then U meets each block in finite number of points.
Theorem 10 is proved.
3. Comments and open questions. We say that an ideal I on a group G is invariant if, for each
A ∈ I
⋃
g∈G g
−1Ag ∈ I. If X is a left-regular G-space and I is an invariant group ideal on G then,
the mapping
B(G,X, I)× B(G,X, I)→ B(G,X, I),
(g, x) 7→ gx is a ≺-mapping.
Question 1. Given a ballean B on the set X, how to detect whether B is asymorphic to
B(G,X, I) for some subgroup G of SX and some invariant group ideal I on G?
A subset A of a topological group G is called bounded if, for every neighbourhood U of the
identity, there exists a finite subset F of G such that A ⊆ FU, A ⊆ UF. The family I of all bounded
subsets of G forms an invariant group ideal. For balleans on topological groups determined by these
ideals see [1].
Question 2. Given an invariant group ideal I on a group G, how to detect whether there is a
group topology τ on G such that I is an ideal of bounded subsets of the topological group (G, τ)?
We say that a filter ϕ on a group G is a group filter if, for any B ∈ ϕ, there exists A ∈ ϕ such
that A = A−1 and AA ⊆ B. A filter ϕ is invariant if, for any B ∈ ϕ and g ∈ G, there is A ∈ ϕ such
that g−1Ag ⊆ B. Clearly, each invariant group filter is a base at identity for some group topology
on G.
Let X be a G-space, ϕ be a group filter on G. The triple (G,X,ϕ) determines a uniformity U on
X with a base of entourages of the diagonal ∆X consisting of all subsets of the form {(x, y)G : y ∈
∈ Ax}, A ∈ ϕ. If ϕ is invariant then, for each g ∈ G, the mapping X → X, x 7→ gx is uniformly
continuous in U .
Question 3. Given a uniform space (X,U), how to detect whether there exist a subgroup G of
SX and a group ideal (an invariant group ideal) I such that the triple (G,X,ϕ) determines (X,U)?
In the following questions “T -point” means “T -point in the class of all metrizable uniformly
locally finite balleans on ω”.
An ultrafilter U ∈ ω∗ is a P -point if and only if, for every Hausdorff topology τ on ω, some
member U ∈ U has at most one limit point in (ω, τ), in particular, some member of U is discrete in
(ω, τ).
Question 4 (T. Banakh). Let U be a free ultrafilter on ω such that, for each metrizable topology
τ on ω, some member of U is discrete. Is U a T -point?
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
350 O. V. PETRENKO, I. V. PROTASOV
We say that an ultrafilter U ∈ ω∗ is sequentially selective if, for any sequence (Pn)n∈ω of
uniformly bounded partitions of ω, there is U ∈ U such that, for each n ∈ ω, |P ∩U | 6 1 for all but
finitely many P ∈ Pn. By Theorem 9, each T -point is sequentially selective.
Question 5. Is every sequentially selective ultrafilter U ∈ ω∗ a T -point ? Does there exist a
sequentially cell selective ultrafilter in ZFC ?
An ultrafilter U ∈ ω∗ is called rapid if, for any partition {PnG : n ∈ ω} of ω into finite subsets,
there exists U ∈ U such that |U ∩ Pn| 6 n for each n ∈ ω.
Jana Flašková noticed that a rapid ultrafilter needs not to be a T -point. Her arguments: a square
of rapid ultrafilters is rapid but a product of two free ultrafilters could not be a T -point. If U , V are
ultrafilters on ω then UV is an ultrafilter on ω×ω with the base of subsets of the form
⋃
x∈U (x, Vx),
Vx ∈ V, U ∈ U . To see that UV is not a T -point, we can either apply Theorem 8 or endow ω with
the structure of an arbitrary group and use the definition of a T -point.
Each countable group G of permutations of ω is contained in some 2-generated subgroup of Sω
(see [2]), so in the definition of a T -point given in [3] we can use only 2-generated subgroup of Sω.
We say that an ultrafilter U ∈ ω∗ is a cyclic T -point if, for each infinite cyclic subgroup G of Sω,
there exists U ∈ U thin in the ballean B(G,ω,Fg).
Question 6. Is every cyclic T -point a T -point ?
Let B = (X,P,B) be a ballean. A subset L of X is called large if there exists α ∈ P such that
X = B(x, α). A subset S of X is called small if (X \ S) ∩ L is large for each large subset L.
Let G be a group of all permutations of ω with finite support (supp g = {x ∈ ωG : gx 6= x}). A
subset A of ω is small in B(G,ω,FG) if and only if A is finite, but each subset of ω is thin. On the
other hand, if X is left regular G-space then each thin subset is small.
Question 7. Does there exist a ZFC-example of a free ultrafilter on ω such that, for every
countable group G and every left regular action of G on ω, there is a member of U small in the
ballean B(G,ω,FG)? Is a weak P -point such an ultrafilter ?
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Received 31.08.11
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
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| id | umjimathkievua-article-2580 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:26:10Z |
| publishDate | 2012 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f6/7da38e66b508ad57971cfd32716b13f6.pdf |
| spelling | umjimathkievua-article-25802020-03-18T19:30:02Z Balleans and G -spaces Болеани та G-простори Petrenko, O. V. Protasov, I. V. Петренко, О. В. Протасов, І. В. We show that every ballean (equivalently, coarse structure) on a set $X$ can be determined by some group $G$ of permutations of $X$ and some group ideal $\mathcal{I}$ on $G$. We refine this characterization for some basic classes of balleans: metrizable, cellular, graph, locally finite, and uniformly locally finite. Then we show that a free ultrafilter $\mathcal{U}$ on $\omega$ is a $T$-point with respect to the class of all metrizable locally finite balleans on $\omega$ if and only if $\mathcal{U}$ is a $Q$-point. The paper is concluded with а list of open questions. Доведено, що кожен болеан (еквiвалентно, груба структура) на множинi $X$ може бути визначений деякою групою пiдстановок $G$ множини $X$ та деяким груповим iдеалом $\mathcal{I}$ на $G$. Цю характеризацiю уточнено для деяких основних класiв болеанiв: метризовних, стiльникових, графових, локально скiнченних, рiвномiрно локально скiнченних. Далi ми доводимо, що вiльний ультрафiльтр $\mathcal{U}$ на $\omega$ є $T$-точкою вiдносно класу метризовних локально скiнченних болеанiв на $\omega$ тодi i тiльки тодi, коли $\mathcal{U}$ є $Q$-точкою. Насамкiнець наведено список вiдкритих проблем. Institute of Mathematics, NAS of Ukraine 2012-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2580 Ukrains’kyi Matematychnyi Zhurnal; Vol. 64 No. 3 (2012); 344-350 Український математичний журнал; Том 64 № 3 (2012); 344-350 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2580/1919 https://umj.imath.kiev.ua/index.php/umj/article/view/2580/1920 Copyright (c) 2012 Petrenko O. V.; Protasov I. V. |
| spellingShingle | Petrenko, O. V. Protasov, I. V. Петренко, О. В. Протасов, І. В. Balleans and G -spaces |
| title | Balleans and G -spaces |
| title_alt | Болеани та G-простори |
| title_full | Balleans and G -spaces |
| title_fullStr | Balleans and G -spaces |
| title_full_unstemmed | Balleans and G -spaces |
| title_short | Balleans and G -spaces |
| title_sort | balleans and g -spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2580 |
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