The connection matrix theory for semiflows on (not necessarily locally compact) metric spaces

The index theory of Rybakowski for isolated invariant sets and attractor-repeller pairs in the setting of a semiflow on a not necessarily locally compact metric space is extended to include a connection matrix theory for Morse decompositions. Partially ordered Morse decompositions and attractor semifiltrations of invariant sets are defined and shown to be equivalent. The definition and proof of existence of index filtrations for an ordered Morse decomposition is provided. Via the index filtration, the homology index braid and the connection matrices of the Morse decomposition are defined.