Preservation and reflection of size properties of balleans

Abstract

We study the combinatorial size of subsets of a ballean, as defined in [19,23] (largeness, smallness, extralargeness, etc.), paying particular attention to the preservation of these properties under taking images and inverse images along various classes of maps (bornologous, effectively proper, (weakly) soft, coarse embeddings, canonical projections of products, canonical inclusions of co-products, etc.). We show by appropriate examples that many of the properties describing the size are not preserved under coarse equivalences (even injective or surjective ones), whereas largeness and smallness are preserved under arbitrary coarse equivalences

Balleans, hyperballeans and ideals

“…Moreover, B ♭ is a subballean of B M . This motivation for the choice of M comes from the fact that non-thick subsets 3 were called meshy in [5] (this term will not be adopted here). (v) the map C : X → P(X) is an asymorphism between X and C(X); (vi) every function f : X → {0, 1} is slowly oscillating.…”

The normality and bounded growth of balleans

“…A family I of subsets of a set X is called an ideal on X if I is closed under finite unions and taking subsets, and X / ∈ I. More information on balleans and coarse spaces can be found in the monographs [4], [14], [18], [20], [21] and in the papers [6], [7], [8], [9], [13].…”

“…B and the preservation of the normality by taking subballeans. The condition (6) implies (2) ∨ (4) ∨ (7).…”

Nonstandard methods in large-scale topology

“…In the light of the topology of * X defined by Corollary 1.4, small subsets do not precisely correspond to nowhere dense subsets: if * A is nowhere dense, then * A ⊆ G c X ( * A) = C c X (G c X ( * A)) = ∅ by Theorem 1.3, so * A must be empty; however, every unbounded connected coarse space has a non-empty small subset. [7,Theorem 2.14]). Let X be a non-empty connected coarse space.…”