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Orbitally discrete coarse spaces. (English) Zbl 1502.54010

The paper under review clarifies the relationship among (orbitally) discrete, (almost) finitary, and scattered coarse spaces.
A coarse space \((X, \mathcal{E})\) is said to be discrete if for each \(E \in \mathcal{E}\) there exists a bounded subset \(B\) such that \(E[x]=\left\{x\right\}\) for each \(x \in X\setminus B\). Intuitively, \(X\) is discrete if it looks discrete from an (ideal) infinitely far point. This property is also known as thin [I. Lutsenko and I. Protasov, Appl. Gen. Topol. 11, No. 2, 89–93 (2010; Zbl 1207.54010)] or pseudodiscrete [I. V. Protasov, Mat. Stud. 20, No. 1, 3–16 (2003; Zbl 1053.54503)].
Given a coarse space \((X, \mathcal{E})\) endowed with the discrete topology, let \(X^{\sharp}\) be the set of all ultrafilters \(p\) on \(X\) such that each member of \(p\) is unbounded in \(X\). Two ultrafilters \(p,q \in X^{\sharp}\) are said to be parallel (\(p \parallel q\)) if there exists an \(E \in \mathcal{E}\) such that \(E[P] \in q\) for each \(P\in p\). or in other words, \(p\) and \(q\) look the same from an infinitely far point. The equivalence classes \(\bar{\bar{p}}\) (\(p\in X^{\sharp}\)) modulo \(\parallel\) are called orbits. The space \(X\) is said to be orbitally discrete if all orbits are (topologically) discrete in the Stone-Čech compactification \(\beta X\). Every discrete space \(X\) is orbitally discrete. See [Protasov, loc. cit.] for more details.
A coarse space \((X, \mathcal{E})\) is said to be finitary if for each \(E \in \mathcal{E}\) there exists a natural number \(n\) such that \(\left|E[x]\right| < n\) for each \(x \in E\). On the other hand, \(X\) is said to be almost finitary if for each \(E \in \mathcal{E}\) there exist a natural number \(n \in \omega\) and a bounded subset \(B\) such that \(\left|E[x]\right| < n\) for each \(x \in E \setminus B\). Clearly all finitary spaces and discrete spaces are almost finitary.
A coarse space \((X, \mathcal{E})\) is said to be scattered if for every unbounded subset \(A\) there exists an \(E\in \mathcal{E}\) such that for each \(E'\in \mathcal{E}\) there is an \(a\in A\) such that \(E_{A}'[a] \setminus E_{A}[a] = \varnothing\). It is an asymptotic analogue of a scattered topological space: the condition \({E'}_{A}[a] \setminus E_{A}[a] = \varnothing\) means that the coarse point \(E_{A}[a]\) is coarsely isolated in \(A\) (i.e. its \(E'\)-neighbourhood has no point other than \(E_{A}[a]\)).
The author first proves that every orbitally discrete space is almost finitary (Theorem 2.1). This theorem follows from the fact that \(X^{\sharp}\) is a closed (hence compact) subset of \(\beta X\).
To further investigate the structure of an almost finitary space, the author employs the construction of a strengthening (ballean-ideal mix) introduced in [O. V. Petrenko and I. V. Protasov, Mat. Stud. 38, No. 1, 3–11 (2012; Zbl 1301.05350)]. A bornology \(\mathcal{B}\) on (the underlying set of) a coarse space \((X, \mathcal{E})\) is said to be \(\mathcal{E}\)-compatible if \(\mathcal{B}\) is closed under taking \(E\)-neighbourhoods for all \(E\in \mathcal{E}\). In such a case, the \(\mathcal{B}\)-strengthening of \((X, \mathcal{E})\) can be defined as the coarse space \((X, \mathcal{H})\) where the bornology associated with \(\mathcal{H}\) is enlarged by \(\mathcal{B}\).
The author then proves that for a coarse space \((X, \mathcal{E})\) the following conditions are equivalent (Theorem 2.2): \((X, \mathcal{E})\) is almost finitary; it is the \(\mathcal{B}\)-strengthening of some finitary coarse space \((X, \mathcal{E}')\), where \(\mathcal{B}\) is the bornology induced by \(\mathcal{E}\); and \(\mathcal{E}\) is the upper bound of a discrete and a finitary coarse structure on \(X\).
Using the characterisation theorem for an almost finitary space to be scattered, the author shows that every orbitally discrete space is scattered (Theorem 2.6). Hence every orbitally discrete space is almost finitary and scattered. Finally the author asks whether the converse is true.

MSC:

54D80 Special constructions of topological spaces (spaces of ultrafilters, etc.)
20B35 Subgroups of symmetric groups
20F69 Asymptotic properties of groups

References:

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