Coarse topology (synonym: coarse geometry) is a branch of geometry/topology studying the asymptotic properties at large of metric spaces. While the topological structures describe the space’s local properties, the coarse structure describes the properties at large distances. One usually studies coarse structure by defining a “coarse category” of proper metric spaces where the morphisms are so called coarse maps.
Do not confuse the field of coarse topology with a comparative property of one topological structure on a set to be coarser than another which is then called finer. Finer or stronger topology has “more” open sets than coarser or weaker – this is of course in the sense of the partial order with respect to inclusion of the topological structures considered as families of subsets.
The field grew in part from the study of “wild” metric structures in study of Gromov hyperbolic groups, from attacks on Novikov conjecture and the related Baum-Connes conjecture. Thus the operator algebras play a great role in the field.
Related Lab entries: asymptotic dimension
A short survey on coarse topology, pdf
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B. Hanke, D. Kotschick, J. Roe, T. Schick, Coarse topology, enlargeability, and essentialness, arxiv:0707.1999
Mikhail Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ. 8, Springer, New York, 1987, pp. 75–263. MR MR919829 (89e:20070) 61; Asymptotic invariants of infinite groups, in: Geometric group theory, vol. 2 (Sussex, 1991), London Math. Soc. Lecture Note Ser. 182, Cambridge Univ. Press 1993, pp. 1–295. MR1253544 (95m:20041)
B. Grave, Asymptotic dimension of coarse spaces, New York J. Math. 12 (2006) 249–256