  In this paper we introduce a notion of an attractor for local semiflows on
topological spaces, which in some cases seems to be more suitable than the
existing ones in the literature. Based on this notion we develop a basic
attractor theory on topological spaces under appropriate separation axioms.
First, we discuss fundamental properties of attractors such as maximality and
stability and establish some existence results. Then, we give a converse
Lyapunov theorem. Finally, the Morse decomposition of attractors is also
addressed.
