Let O1, . . . , On be open sets in C[0, 1], the space of real-valued continuous functions on [0, 1]. The product O1 · · · On will in general not be open, and in order to understand when this can happen we study the following problem: given f1, . . . , fn ∈ C[0, 1], when is it true that f1 · · · fn lies in the interior of Bε(f1) · · · Bε(fn) for all ε > 0 ? (Bε denotes the closed ball with radius ε and centre f .) The main result of this paper is a characterization in terms of the walk t → γ(t)

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... = (f1(t), . . . , fn(t)) in R n . It has to behave in a certain admissible way when approaching {x ∈ R n | x1 · · · xn = 0}. We will also show that in the case of complex-valued continuous functions on [0, 1] products of open subsets are always open. 2010 Mathematics Subject Classification: 46B20, 46E15.