The notion of a bisimulation relation is of
basic importance in many areas of computation theory and logic.
Of late, it has come to take a particular significance in work
on the formal analysis and verification of hybrid control systems,
where system properties are expressible by formulas of the modal μ-calculus or
weaker temporal logics. Our purpose here is to give an analysis of the
concept of bisimulation, starting with the observation that the zig-zag
conditions are suggestive of some form of continuity. We give a topological
characterization of bisimularity for preorders, and then use the topology
as a route to examining the algebraic semantics for the µ-calculus,
developed
in recent work of Kwiatkowska et al., and its relation to the standard
set-theoretic semantics. In our setting, μ-calculus sentences
evaluate as clopen sets of an Alexandroff topology, rather than as clopens
of a (compact, Hausdorff) Stone topology, as arises in the Stone space representation
of Boolean algebras (with operators). The paper concludes by applying the topological
characterization to obtain the decidability of μ-calculus properties for a
class of first-order definable hybrid dynamical
systems, slightly extending and considerably simplifying the proof of a recent
result of Lafferriere et al.