Ultrafilters on Balleans

On a question of Dikranjan and Zava

“…We denote by Φ the family of all mappings φ : ω −→ [ω] <ω such that, for each n ∈ ω, n ∈ φ(n) and {m : n ∈ φ(m)} is finite. We consider the family F of all subsets of ω × ω of the form {(n, k) : k ∈ φ(n), n ∈ ω}, φ ∈ Φ and note that F is a coarse structure on ω, see [11,Example 1.4.6]. The universal property of (ω, F ) : if E is a coarse structure on ω such that every bounded subset in (ω, E) is finite then each injective mapping f : (ω, E) −→ (ω, F ) is macro-uniform.…”

“…Now let G be a group. A family I of subsets of G is called a group ideal [9], [11] if G contains the family [G] <ω of all finite subsets of G and A, B ∈ I, C ⊆ A imply AB −1 ∈ I, C ∈ I. Every group ideal I defines a coarse structure on G with the base {{(x, y) : x ∈ Ay} ∪ △ G : A ∈ I}.…”

Sequential coarse structures of topological groups

“…We define an equivalence ∼ on X ♯ by the rule: p ∼ q if and only if f β (p) = f β (q) for every slowly oscillating function f : X −→ {0, 1}. The quotient X ♯ / ∼ is called a space of ends or binary corona of (X, E), see [12,Chapter 8].…”

“…Now let G be a group. A family I of subsets of G is called a group ideal [10], [12] if G contains the family [G] <ω of all finite subsets of G and A, B ∈ I, C ⊆ A imply AB −1 ∈ I, C ∈ I. Every group ideal I defines a coarse structure on G with the base {{(x, y) : x ∈ Ay} : A ∈ I}.…”

Closeness and linkness in balleans

“…The pair (X, E) is called a coarse space [13] or a ballean [10], [12]. For a ballean (X, E), a subset B ⊆ X is called bounded if B ⊆ E[x] for some E ∈ E and x ∈ X.…”