Let $G$ be a Lie group.

In the book Curvature and Characteristic classes, the author (Johan L. Dupont) mentiones in beginning of chapter 5 the following :

The notion of a topological principal $G$-bundle $\pi:E\rightarrow X$ on a topological space $X$ is defined exactly as in Definition $3.1$ (the definition of usual principal bundle over manifold), only the words "differentiable" and "diffeomorphism" are replaced by "continuous" and "homomorphism". The purpose of this and the following section is to show that the Chern-Weil morphism defines characteristic classes of topological G-bundles.

But does not mention (or I could not find) anything about characteristic classes of topological principal bundles. Are there other places that discuss the notion of characteristic classe of topological principal bundles using Chern-Weil theory?