  We define E-theory for separable C*-algebras over second countable
topological spaces and establish its basic properties. This includes an
approximation theorem that relates the E-theory over a general space to the
E-theories over finite approximations to this space. We obtain effective
criteria for determining the invertibility of E-theory elements over possibly
infinite-dimensional spaces. Furthermore, we prove a Universal Multicoefficient
Theorem for C*-algebras over totally disconnected metrisable compact spaces.
