Coarse amenability and discreteness (Q2788670)

Coarse geometry studies asymptotic properties of spaces. An important source of ideas comes from the dualization of concepts from classical topology to the setting of coarse geometry. While open covers often get replaced by uniformly bounded covers, the dualization of other concepts usually requires a more subtle approach. In his previous work, the author developed a dualization of partitions of unity and successfully applied it to the well established concepts, introducing the concept of a large-scale paracompact space along the way.NEWLINENEWLINEIn this paper the author develops another dualization of paracompactness using the dualization of disjointness in terms of \(R\)-disjointness. The obtained notion, called the countable asymptotic dimension, is a generalization of the straight finite decomposition complexity. The author proves that the obtained class of spaces has Property A. Furthermore, he uses the large scale absolute extensors to provide a condition, under which such spaces are of finite asymptotic dimension.