Topological coarse shape groups of compact metric spaces

Abstract

Abstract. The shape theory and, relatively new, coarse shape theory are very useful in studying of topological spaces, as well as of the corresponding algebraic invariants, especially, shape and coarse shape groups. By using certain ultrametrics on special sets of pro-and pro * -morphisms, we topologize those groups when they refer to compact metric spaces and we get topological groups. In the shape case, they are isomorphic to recently constructed topological shape homotopy groups, while in the coarse shape case we get the coarse shape invariants, denoted byπ * d * k (X, x 0 ). We have proven some important properties ofπ * d * k (X, x 0 ) and provided few interesting examples.