Topological properties of attractors for dynamical systems

For a dynamical system {St} on a metric space X, we examine the question whether the topological properties of X are inherited by the global attractor A (if it exists). When {St} is jointly continuous, we prove that the Čech–Alexander–Spanier cohomology groups of A are isomorphic to the corresponding cohomology groups of X. The same conclusion is obtained in the case where {St} is a group and A has a bounded neighborhood which is a deformation retract of X.