Strong Conley index over a phase space

We extend the definition of the strong Conley index of a discrete dynamical system introduced in [1] to the strong Conley index over a phase space of a discrete dynamical system which strengthens the Conley index over a phase space defined in [2] and also generalizes the strong Conley index of a discrete dynamical system. We also prove the continuation property of the strong Conley index over a phase space.