Conley index theory without index pairs. I: The point-set level theory

Abstract: We propose a new framework for Conley index theory. The main feature of our approach is that we do not use the notion of index pairs. We introduce, instead, the notions of compactifiable subsets and index neighbourhoods, and formulate and prove basic results in Conley index theory using these notions. We treat both the discrete time case and the continuous time case.