Generalization of Stone-Weierstrass for Tychonoff Spaces A well-known generalization of the Stone-Weierstrass Theorem is the following:
Theorem: Suppose $X$ is a Tychonoff space. If $A\subset C(X,\mathbb{R})=\{f:X\to\mathbb{R} \mid f \text{ is continuous}\}$ is a subalgebra which separates points and contains the constant functions, then $A$ is dense in $C(X,\mathbb{R})$ with respect to the compact-open topology.
This result can be found here (a previous question on this website), and also in the introduction to the Wikipedia article about the Stone-Weierstrass Theorem, here
My questions are as follows:


*

*Can someone direct me to a proof of this generalization?

*Can the hypothesis that $A$ contains the constant functions be replaced by the hypothesis that $A$ vanishes at no point of $X$?


The author of the answer to the previous question I referenced on this website mentions that the proofs he or she cites are mostly based on Section 44 of Willard's General Topology. I looked at that section and found that the above theorem and other generalizations of Stone-Weierstrass are given as exercises so their proofs are not included.
Thank you in advance for any help!
 A: Kelley (General Topology, p. 244/245 in exercise R of the chapter on function spaces) formulates it as follows:

If $X$ is a topological space and the family $C(X)$ of all continuous real-valued functions on $X$ is given the topology of uniform convergence on compacta (which is the compact-open topology according to Thm. 11 in the same chapter), then each subalgebra of $C(X)$ that has the two-point property is dense in $C(X)$. 

(the two-point property being that for each $x \neq y$ in $X$ and any two reals $a,b$  there is an $f \in C(X)$ with $f(x) =a \land f(y)=b$; it's implied by all constant functions being in the subalgebra plus the subalgebra separating points, according to the exercise's preamble, and that is indeed easy to see, and baby Rudin shows that nowhere vanishing plus separating points together also imply it; this shows that you don't always need all constant functions to be in the subalgebra). Having constants + nowhere vanishing together might not imply it, though, not sure..
He refers to 
M.H. Stone The generalized Weierstrass approximation theorem, Math. Mag. 21 (1948) 167-184, 254-273
for further discussion (so I presume there's a proof or reference to it there).
I couldn't find a similarly broad theorem in Engelking (usually my go-to book), though he refers to the same paper, as Willard does too BTW (!)
The encyclopedia of general topology's chapter on function spaces mentions (p151 section7, almost as an aside) that the theorem holds for general spaces and subalgebras with constant functions and separating points (so implying the two-point property again) in the compact-open topology, but gives no explicit reference for it. The three general references it does give might help, though. 
Maybe the proof reduces to the compact case (for which Engelking has a full proof, and Kelly extensive hints) in some way? I'd look for the MH Stone paper and see what it has. It might be the original reference for all of this. (it isn't called the Stone-Weierstrass theorem for nothing: Weierstrass did the special case of polynomaials on a compact real interval, nothing as general as Stone's contribution).
