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

Ehrhard Behrends¹

¹ Mathematisches Institut Freie Universität Berlin Arnimallee 6 D-14195 Berlin, Germany

Studia Mathematica, Tome 204 (2011) no. 1, pp. 73-95

Résumé

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

Détail

DOI : 10.4064/sm204-1-5

Keywords: ldots sets space real valued continuous functions product cdots general order understand happen study following problem given ldots cdots lies interior varepsilon cdots varepsilon varepsilon varepsilon denotes closed ball radius varepsilon centre main result paper characterization terms walk mapsto gamma ldots mathbb has behave certain admissible approaching mathbb mid cdots complex valued continuous functions products subsets always open