X is a "nice" topological space (insert whatever definition of nice you like). D and E are closed subsets of X, and I have a function f:X-->R which is zero on (D union E) and positive everywhere else. The problem is to find functions g and h such that g is zero on D, h is zero on E, and 0 < gh < f everywhere else. I've tried working on this problem before, and got nowhere. Been working on it again, and still failing. It's very annoying. More general version: can this be done for any topological space? In case you're curious, this problem came from trying to look at the prime spectrum of the ring of continuous, real-valued functions on X. I conjecture to each prime ideal of C(X), the filter comprised of the zero sets of the functions in the prime ideal is an ultrafilter.