Closeness and linkness in balleans

Abstract

A set X endowed with a coarse structure is called ballean or coarse space. For a ballean (X, E), we say that two subsets A, B of X are close (linked) if there exists an entourage E ∈ E such that A ⊆ E[B], B ⊆ E[A] (either A, B are bounded or contain unbounded close subsets). We explore the following general question: which information about a ballean is contained and can be extracted from the relations of closeness and linkness. 1991 MSC: 54E99, 54D80.

Small uncountable cardinals in large-scale topology

“…On the other hand, we do not know if an inseparated cellular finitary coarse structure on ω exists in ZFC. Corollary 5.10 yields a (consistent) negative answer to Question 6 of Protasov and Protasova [19].…”

“…In Section 6 we use the equality b = c for constructing continuum many large finitary coarse structures on ω, which answers Question 4 of Protasov and Protasova [19] in negative (at least under the assumption b = c).…”

Set-Theoretical Problems in Asymptology

“…• λ-equivalent if two subsets A, B ⊆ X are E-separated if and only if they are E ′separated. These equivalences were introduced and studied in [34] where the following two problems are posed.…”

“…It can be shown that two metrizable coarse structures on the same set are equal if and only if they are δ-equivalent. Problem 2.13 ([24], [34]). Let E, E ′ be two δ-equivalent finitary coarse structures on ω.…”