I am reading a paper which has some complicated construction on a Hausdorff topological space called $C$-distinguished topological space. The paper says that a $C$-distinguished topological space $X$ is a Hausdorff topological space such that the space of real-valued bounded continuous function on $X$, $C_b(X)$, separates points of $X$.
But later it says that most "normal" space we learned so far is usually not a $C$-distinguished topological space.
I confused here. Since I think this definition is not that strong... Anyway, can anybody help me to confirm that at least the usual space $\mathbb R^N$ with usual topology is a $C$-distinguished space or not?