Computing connectedness: An exercise in computational topology

We reformulate the notion of connectedness for compact metric spaces in a manner that may be implemented computationally. In particular, our techniques can distinguish between sets that are connected; have a nite number of connected components; have in nitely many connected components; or are totally disconnected. We hope that this approach will prove useful for studying structures in the phase space of dynamical systems. AMS classication scheme numbers: 54D05, 58F13, 68U05