A motivation: The classical Stone-Weierstrass theorem says that polynomials are dense among continuous functions (say, on the unit interval), while the abstract Stone-Weierstrass theorem (and also the related Kakutani-Krein theorem) give sufficient conditions for a set of continuous functions to be dense.

The classical partitions of unity are continuous ones or smooth ones. I ask if there are abstract theorems on the existence of partitions of unity generalizing the classical ones. Specifically, I'm looking for "known" or "natural" statements of the following form: If a topological space $X$ satisfies [yadda yadda] and a set $F$ of real-valued functions on $X$ satisfies [blah blah] then any open cover of $X$ has a subordinate partition of unity composed of functions in $F$.

As an example of such a statement, replace [yadda yadda] with [locally compact Hausdorff] and [blah blah] with [$F$ is an algebra of continuous functions that satisfies Urysohn's Lemma]. (Proof: the same as in Rudin's RCA book, page 40.) Is there a cleaner formulation?

Related questions: Is there such an "abstract" version of Urysohn's Lemma? Of Tietze Extension Theorem?

Please provide references, if available. :)