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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?

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    $\begingroup$ If a $C$-distinguished space is a Hausdorff space $X$ on which $C_b(X)$ separates points, then every completely regular space ($T_3$ and $T_0$) is $C$-distinguished. In particular, any normal ($T_4$ and $T_1$) space. $\endgroup$ – Daniel Fischer Jul 31 '15 at 12:15
  • $\begingroup$ @DanielFischer I see. So indeed $\mathbb R^N$ is $C$-distinguished right? It looks to me pretty every normal space we met is at least a $T_3$ and $T_0$... $\endgroup$ – spatially Jul 31 '15 at 12:18
  • $\begingroup$ If you haven't misunderstood the definition, it is. If you are using "normal" in the sense "ordinary, typical", then indeed most spaces one encounters in analysis are $C$-distinguished. If you are using "normal" in the topological sense, then depending on the used nomenclature, either all normal spaces are $C$-distinguished, or all normal Hausdorff spaces are $C$-distinguished [there are two common nomenclatures, a) normal = $T_4 + T_1$, and b) $T_4$ = normal $+ \; T_1$]. $\endgroup$ – Daniel Fischer Jul 31 '15 at 12:29
  • $\begingroup$ @DanielFischer I indeed just copied the definition from that paper so it shouldn't be wrong. By normal I mean "ordinary, typical", sorry for confusion! and thx for your comment here. $\endgroup$ – spatially Jul 31 '15 at 12:31
  • $\begingroup$ Such spaces are usually called completely Hausdorff or functionally Hausdorff. $\endgroup$ – Brian M. Scott Aug 1 '15 at 18:23

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