Products of $n$ open subsets in the space of continuous functions on $[0,1]$
Volume 204 / 2011
Abstract
Let $O_{1},\ldots ,O_{n}$ be open sets in $C[0,1]$, the space of real-valued continuous functions on $[0,1]$. The product $O_{1}\cdots O_{n}$ will in general not be open, and in order to understand when this can happen we study the following problem: given $f_{1},\ldots ,f_{n}\in C[0,1]$, when is it true that $f_{1}\cdots f_{n}$ lies in the interior of $B_{\varepsilon}(f_{1})\cdots B_{\varepsilon}(f_{n})$ for all $\varepsilon>0\,$? ($B_{\varepsilon}$ denotes the closed ball with radius $\varepsilon$ and centre $f$.) The main result of this paper is a characterization in terms of the walk $t\mapsto \gamma(t):=(f_{1}(t),\ldots ,f_{n}(t))$ in $\mathbb R^n$. It has to behave in a certain admissible way when approaching $\{x\in\mathbb R^n\mid x_{1}\cdots x_{n}=0\}$. We will also show that in the case of complex-valued continuous functions on $[0,1]$ products of open subsets are always open
