@ Appl. Gen. Topol. 22, no. 2 (2021), 303-309doi:10.4995/agt.2021.13874
© AGT, UPV, 2021

Orbitally discrete coarse spaces

Igor Protasov

Taras Shevchenko National University of Kyiv, Department of Computer Science and Cybernetics,

Academic Glushkov pr. 4d, 03680 Kyiv, Ukraine (i.v.protasov@gmail.com)

Communicated by F. Mynard

Abstract

Given a coarse space (X, E), we endow X with the discrete topology

and denote X
♯ = {p ∈ βG : each member P ∈ p is unbounded }. For

p, q ∈ X♯, p||q means that there exists an entourage E ∈ E such that
E[P ] ∈ q for each P ∈ p. We say that (X, E) is orbitally discrete if, for

every p ∈ X♯, the orbit p = {q ∈ X♯ : p||q} is discrete in βG. We prove
that every orbitally discrete space is almost finitary and scattered.

2010 MSC: 54D80; 20B35; 20F69.

Keywords: coarse space; ultrafilter; orbitally discrete space; almost finitary

space; scattered space.

1. Introduction and preiminaries

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 ∈ 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;

•
⋃

E = X × X.

Received 16 June 2020 – Accepted 15 April 2021

http://dx.doi.org/10.4995/agt.2021.13874


I. Protasov

A subfamily E′ ⊆ E is called a base for E if, for every E ∈ E, there exists
E′ ∈ E′ such that E ⊆ E′. For x ∈ X, A ⊆ X and E ∈ E, we denote

E[x] = {y ∈ X : (x, y) ∈ E}, E[A] =
⋃

a∈A

E[a], EA[x] = E[x] ∩ A

and say that E[x] and E[A] are balls of radius E around x and A.

The pair (X, E) is called a coarse space [19] or a ballean [12], [18].
For a coarse space (X, E), a subset B ⊆ X is called bounded if B ⊆ E[x] for

some E ∈ E and x ∈ X. The family B(X,E) of all bounded subsets of (X, E)
is called the bornology of (X, E). We recall that a family B of subsets of a set
X is a bornology if B is closed under taking subsets and finite unions, and B
contains all finite subsets of X.

A coarse space (X, E) is called finitary, if for each E ∈ E there exists a
natural number n such that |E[x]| < n for each x ∈ X.

Let G be a transitive group of permutations of a set X. We denote by XG
the set X endowed with the coarse structure with the base

{{(x, gx) : g ∈ F}} : F ∈ [G]<ω, id ∈ F}.

By [8, Theorem 1], for every finitary coarse structure (X, E), there exists
a transitive group G of permutations of X such that (X, E) = XG. For more
general results, see [10].

Let X be a discrete space and let βX denote 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, so X∗ = βX \ X is the set of
all free ultrafilters. The topology of βX is generated by the base consisting of
the sets Ā = {p ∈ βX : A ∈ p}, where A ⊆ X. The universal property of βX
states that every mapping f : X −→ Y to a compact Hausdorff space Y can
be extended to a continuous mapping fβ : βX −→ X.

Given a coarse space (X, E), we endow X with the discrete topology and
denote by X♯ the set of all ultrafilters p on X such that each member P ∈ p
is unbounded. Clearly, X♯ is a closed subset of X∗ and X♯ = X∗ if (X, E) is
finitary.

Following [7], we say that two ultrafilters p, q ∈ X♯ are parallel (and write
p||q) if there exists E ∈ E such that E[P ] ∈ q for each P ∈ p. Then || is an
equivalence on X♯. We denote

p = {q ∈ X♯ : q||p}

and say that p is the orbit of p. If (X, E) is finitary and (X, E) = XG then
p = Gp.

A coarse space (X, E) is called orbitally discrete if, for every p ∈ X♯, the
orbit p is discrete. Every discrete coarse space is orbitally discrete. We recall
that (X, E) is discrete if, for each E ∈ E, there exists a bounded subset B such
that E[x] = {x} for each x ∈ X \ B. In this case, p = {p} for each p ∈ X♯.

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Orbitally discrete coarse spaces

Every bornology B on a set X defines the discrete coarse structure on X with
the base {EB : B ∈ B}, EB[x] = B if x ∈ B, and EB[x] = {x} if x ∈ X \ B.

By [15, Theorem 5.4], for a finitary coarse space (X, E), the following con-
ditions are equivalent: XG is orbitally discrete, XG is scattered, XG has no
piecewise shifted FP-sets.

A coarse space (X, E) is called scattered if, for every unbounded subset A
of X, there exists E ∈ E such that A has asymptotically E-isolated balls: for
each E′ ∈ E, there is a ∈ A such that E′A[a] \ EA[a] = ∅.

This notion arouse in the characterization of the Cantor macrocube [3] and,
in the case of finitary coarse groups, was explored in [2].

Let G be a group of permutations of a set X. Let (gn)n∈ω be a sequence in
G and let (xn)n∈ω be a sequence in X such that

(1) {gǫ00 . . . g
ǫn
n xn : (ǫi)

n
i=0 ∈ {0, 1}

n+1} ∩ {gǫ00 . . . g
ǫm
m xm : (ǫi)

n
i=0 ∈

{0, 1}n+1} = ∅ for all distinct n, m ∈ ω;
(2) {gǫ00 . . . g

ǫn
n xn : (ǫi)

n
i=0 ∈ {0, 1}

n+1}| = 2n+1 for every n ∈ ω.

Following [15], we say that a subset Y of X is a piecewise shifted FP-set if
there exist (gn)n∈ω, (xn)n∈ω satisfying (1), (2) and such that

Y = {gǫ00 . . . g
ǫn
n xn : ǫi ∈ {0, 1}}, n ∈ ω}.

After exposition of results in Section 2, we survey some known classes of
orbitally discrete spaces in Section 3.

2. Resuts

A coarse space (X, E) is called almost finitary if, for every E ∈ E , there
exists a bounded subset B and a natural number n such that |E[x]| < n for
each x ∈ X \ B. Every discrete space and every finitary space are almost
finitary.

Theorem 2.1. Every orbitally discrete coarse space is almost finitary.

Proof. We suppose the contrary and choose E ∈ E, E = E−1 such that, for
any bounded subset B and a natural number n, there exists x ∈ X \ B such
that |E[x]| > n.

We claim that there exists p ∈ X♯ such that, for every P ∈ p, {x ∈ P :
|E2[x] ∩ P | > 1} ∈ p. Otherwise, for every p ∈ X♯, there exists Qp ∈ p
such that {x ∈ Qp : E

2[x] ∩ Qp| = 1} ∈ p. We consider the open covering
{Q♯p : p ∈ X

♯} of X♯ and choose its finite subcovering Q♯p1, . . . , Q
♯
pm

. Then the
set B = X\(Qp1∪, . . . , ∪Qpm) is bounded and |E[x]| ≤ m for each x ∈ X\E[B],
but this contradicts the choice of E.

We show that the orbit p is not discrete. Given any P ∈ p, we choose Q ∈ p,
Q ⊆ P such that |E2[x] ∩ P | > 1 for each x ∈ Q. For every x ∈ Q, we take
f(x) ∈ E2[x] ∩ P such that x 6= f(x). Then we extend the mapping x 7→ f(x)

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I. Protasov

from Q to X by f(x) = x for each x ∈ X \ Q. Clearly, fβ(p) 6= p, P ∈ fβ(p)
and fβ(p)||p because (x, f(x)) ∈ E2 for each x ∈ X. �

To clarify the structure of an almost finitary coarse space, we use the fol-
lowing construction from [6]. A bornology B on a coarse space (X, E) is called
E-compatible if E[B] ∈ B for all B ∈ B, E ∈ E. Every E-compatible bornology
B defines the B-strengthening (X, H) of (X, E), where H has the base

{HB,E : B ∈ B, E ∈ E},

HB,E[x] =

{

E[B], if x ∈ B,

E[x], if x ∈ X \ B.

For description of the upper bound E ∨ E′ of coarse structures, see [13].

Theorem 2.2. For a coarse space (X, E), the following statements are equiv-
alent

(i) (X, E) is almost finitary;
(ii) (X, E) is the B-stregthening of some finitary coarse space (X, E′) by the

bornology B of bounded subspaces of (X, E);
(iii) E is the upper bound of a discrete and a finitary coarse structures on

X.

Proof. (i) =⇒ (ii). For B ∈ B and E ∈ E, we pick B′B,E ∈ B and a natural

number n such that B ⊆ B′B,E and |E[x]| < n for each x ∈ X \ B
′
B,E. We note

that {B′B,E : B ∈ B, E ∈ E} is a base for B. For B ∈ B, E ∈ E we put

E′B,E =

{

x if x ∈ B′B,E,

E[x] if x ∈ X \ B′B,E,

denote by E′ the smallest coarse structure on X containing all entourages
{HB,E : B ∈ B, E ∈ E}, observe that E

′ is finitary and (X, E) is the B-
strengthening of (X, E′).

(ii) =⇒ (iii). If (X, E) is the B-strengthening of (X, E′) then E is the upper
bounded of E′ and the discrete coarse structure on X defined by the bornology
B.

(iii) =⇒ (i). We assume that E is the upper bound of finitary coarse struc-
ture E′ and discrete coarse structure on X defined by some bornology B. We
choose the smallest bornology B′ on X such that B ⊆ B′ and E′(B′) ∈ B′ for
all E′ ∈ E′. Then B′ is the bornology of bounded subsets of (X, E) and (X, E)
is the B′-strengthening of (X, E′), so (X, E) is almost finitary. �

Remark. Let (X, E) be the B-strengthening of a finitary coarse space
(X, E′). If (X, E′) is orbitally discrete then (X, E) is orbitally discrete, but
the converse statement needs not to be true. Let X be the disjoint union of

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Orbitally discrete coarse spaces

two infinite subsets Y, Z. We endow Y with the finitary coarse structure EY
such that (Y, EY ) is not orbitaly discrete, and denote by EZ the discrete coarse
structure on Z defined by the bornology of finite subset. We take the smallest
coarse structure E′ on X such that E′|Y = EY , E

′|Z = EZ. Clearly, E
′ is finitary

but not orbitally discrete. We denote by B the smallest bornology on X such
that Y ∈ B. Then the B-strengthening of (X, E′) is discrete.

Theorem 2.3. For almost finitary coarse space (X, E) and p, q ∈ X♯, we have
p||q if and only if there exist E ∈ E and a permutation g of X such that gp = q,
gp = {gP : P ∈ p} and (x, gx) ∈ E for each x ∈ X.

Proof. Let p||q. We take E ∈ E such that E = E−1 and E[P ] ∈ q for each
P ∈ p. Since (X, E) is almost finitary, there exist a bounded subset B of X
and a natural number n such that |E[x]| < n for each x ∈ X \ B. We put
Y = X \ E[B], note that Y ∈ p and define a set-valued mapping F : X −→
[x]<ω. F(x) = E[x] if x ∈ Y and F(x) = {x} if x = X \ Y . By Theorem 1
from [10], there exists bijection f1, . . . , fm of X such that fi(x) ∈ F(x) and
f1(x)∪· · · ∪fn(x) = F(x). We take i ∈ {1, . . . , m} such that fi(P) ∈ q for each
P ∈ p and put g = fi.

The converse statement follows directly from the definition of the parallelity
relation ||. �

Corollary 2.4. If (X, E) is almost finitary, p ∈ X♯ and p is an isolated point
of p then p is discrete.

Proof. We assume that some point q ∈ p is not isolated in p, use Theorem 2.3
to choose a permutation g of X such that gq = p and note that p is not isolated
in p. �

For a subset A of (X, E) and p ∈ X♯, we denote ∆p(A) = p ∩ A
♯.

Theorem 2.5. An almost finitary coarse space (X, E) is scattered if and only
if, for every unbounded subset A of X, there exists p ∈ A such that ∆p(A) is
finite.

Proof. We suppose that X is scattered and choose E ∈ E such that A has an
asymptotically isolated E-balls. For each H ∈ E, we denote PH = {x ∈ A :
HA[x] \ EA[x] = ∅} and take p ∈ A

♯ such that PH ∈ p for each H ∈ E. If
q ∈ A♯ and q||p then E[P ] ∈ q for each P ∈ p. We take the bijections f1, . . . , fm
from the proof of Theorem 2.3. Since q = gp for some g ∈ {f1, . . . , fm}, we
have ∆p(A) ≤ m.

Let ∆p(A) = {p1, . . . , pm}. For each i ∈ {1, . . . , m}, we pick Ei ∈ E such
that Ei[P ] ∈ pi for each P ∈ p. Then we take E ∈ E such that Ei ⊆ E for each
i ∈ {1, . . . , m}, and observe that A has an asymptotically isolated E-balls. �

Theorem 2.6. Every orbitally discrete space is scattered.

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I. Protasov

Proof. To apply Theorem 2.5, we take an arbitrary unbounded subset A of X
and find p ∈ A♯ such that ∆p(A) is finite.

We use the Zorn lemma to choose a minimal (by inclusion) closed subset S
of A♯ such that ∆q(A) ⊆ S for each q ∈ S. Let p ∈ S but ∆p(A) is infinite. We
take the limit point q of ∆p(A). By the minimality of S, we have p ∈ cl∆q(A).
Applying Theorem 2.3, we conclude that p is not isolated in p. �

Question. Let X be an almost finitary scattered space. Is X orbitally discrete?

3. Comments

1. For a natural number n, a coarse space (X, E) is called n-thin if, for
every E ∈ E, there exists a bounded subset B of X such that |E[x]| ≤ n,
for every x ∈ X \ B. A space (X, E) is n-thin if and only if |p| ≤ n for
each p ∈ X♯.

For finite partitions of an n-thin space into discrete subspaces, see
[5], [14], [17], [1, Section 6].

2. A coarse space (X, E) is called sparse if each orbit p, p ∈ X♯ is finite.
Sparse subsets of groups are studied in [4], [16]. For sparse metric
spaces, see [9].

3. A coarse space (X, E) is called indiscrete if each discrete subspace of X
is bounded. By Theorem 3.15 from [11], a finitary indiscrete space has
no unbounded orbitally discrete subspaces. We do not know whether
this statement holds for any almost finitary indiscrete spaces.

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