An n-dimensional CW-complex is a CW-complex admitting a cell structure of (not necessarily finite) cells of maximal dimension n. A topological n-manifold is a locally euclidian (not necessarily compact) Hausdorff space with a countable base.

Let $M=\mathbb{R}$. In topology 4, all functions with compact support whose second derivative is smaller than $1$ would be open. In the Whitney $C^{\infty}$ topology they would'nt since their values and first derivatives must be close too, if one requires something for the second derivative. Am I missing something or does this show topology 4 differs from the Whitney topology?

@DanRamras Thank you, this answers the third question. Can you also say something about the other three?

@OldřichSpáčil Thank you, I managed to prove the result with inspiration from the proof of Theorem 44.1 of 'Kriegl, Michor'. (Which has some flaws by the way...)

@IgorKhavkine That's it, thank you. Do you or somebody else have a reference or even a proof of this result?

The proof of of (4.1) in Segal's "Classifying Spaces and Spectral Sequences" (math.northwestern.edu/~konter/gtrs/segal.pdf) should exemplify my doubts more closely. I don't see the map $\phi\colon X\rightarrow BR_U$ being continuous. Do you or somebody else do?

@OmarAntolín-Camarena That would be an option, but just shows me, that my example is bad chosen. In the situations I have in mind, the things corresponding to the $a_i$'s are only part of the nerve, if the corresponding function of the partition of unity is nonzero. Then one can not proceed as you suggested.