Selectors of discrete coarse spaces
Igor Protasov

TL;DR
This paper characterizes when discrete coarse spaces admit selectors and 2-selectors, linking these properties to the existence of a linear order whose intervals form a base for the space's bornology.
Contribution
It establishes an equivalence between the existence of selectors in discrete coarse spaces and the presence of a compatible linear order with a specific interval base.
Findings
Selectors exist if and only if a compatible linear order exists.
Selectors are equivalent to 2-selectors in discrete coarse spaces.
The interval family from the linear order forms a base for the bornology.
Abstract
Given a coarse space with the bornology of bounded subsets, we extend the coarse structure from to the natural coarse structure on and say that a macro-uniform mapping (resp. ) is a selector (resp. 2-selector) of if for each (resp. . We prove that a discrete coarse space admits a selector if and only if admits a 2-selector if and only if there exists a linear order on such that the family of intervals is a base for the bornology .
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Taxonomy
TopicsAdvanced Topology and Set Theory · Advanced Operator Algebra Research · Advanced Banach Space Theory
