I know (and am able to prove via Stone-Čech compactification) that the following is correct:



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> **Theorem:** A metric space is compact if and only if its space of bounded, continuous, real-valued functions is separable in the uniform topology.
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I use it in a paper for readers who are presumably not familiar with this kind of topology, so I cannot call it "obvious" or "well-known".
I would be thankful for a name and/or good reference to cite this theorem!



reference for "X compact <=> C_b(X) separable" (X metric space)
The result does appear in Dunford/Schwartz, *Linear Operators Part I* (page 437), but is only stated as an exercise. 


Edit after @JosephVanName' comment:
Conway's *Functional Analysis* has the result for completely regular spaces as Theorem 6.6 (page 140).


