Products of n open subsets in the space of continuous functions on [0,1]

Let O-1, ... , O-n be open sets in C[0, 1], the space of real-valued continuous functions on [0, 1]. The product O-1 ... O-n will in general not be open, and in order to understand when this can happen we study the following problem: given f(1), ... , f(n) is an element of C[0, 1], when is it true that f(1) ... f(n) lies in the interior of B-epsilon(f(1)) ... B-epsilon(f(n)) for all epsilon > 0? (B-epsilon denotes the closed ball with radius epsilon and centre f.) The main result of this paper is a characterization in terms of the walk t (sic) gamma(t) := (f(1)(t), ... , f(n)(t)) in R-n. It has to behave in a certain admissible way when approaching {x is an element of R-n vertical bar x(1) ... x(n) = 0}. We will also show that in the case of complex-valued continuous functions on [0, 1] products of open subsets are always open.