Shape and Morse theory of attractors

We study the shape (in the sense of Borsuk) of attractors of continuous semi‐dynamical systems on general metric spaces. We show, in particular, that the natural inclusion of the global attractor into the state space is a shape equivalence. This and other results of the paper are used to develop an elementary Morse theory of an attractor. © 2000 John Wiley & Sons, Inc.