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 :



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> 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.
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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?



Chern -Weil map for topological principal G bundles
I think "Chern–Weil theory" here is being used in a slightly nonstandard way. The main result of the chapter, Theorem 5.5, is really just a topological statement that elements of $H^\*(BG,\Lambda)$ correspond exactly to characteristic classes (Definition 5.1), which are defined to be functorial associations of ($\Lambda$-valued) **singular** cohomology classes to isomorphism classes of topological bundles. This is combined with the calculation that the cohomology of $BG$ is calculated to be a certain invariant polynomial ring (assuming, eg, $G$ is compact), i.e. the [Chern–Weil homomorphism](https://en.wikipedia.org/wiki/Chern%E2%80%93Weil_homomorphism), since the group $G$ is still a Lie group.


In traditional Chern–Weil theory (for smooth bundles on manifolds), one also has the isomorphism of de Rham with singular cohomology, and, most importantly, that the invariant polynomials can be evaluated on curvature forms to give de Rham classes. 


But in the topological case at hand, one still has invariant polynomials giving rise to characteristic classes, which is what I think Dupont means here.


