E-theory for C⁎-algebras over topological spaces

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.