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 for all ε > 0 ? ( 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): = (f₁(t),..., fₙ(t)) in ℝⁿ. It has to behave in a certain admissible way when approaching x ∈ ℝⁿ | 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
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm204-1-5,
author = {Ehrhard Behrends},
title = {Products of n open subsets in the space of continuous functions on [0,1]},
journal = {Studia Mathematica},
volume = {204},
year = {2011},
pages = {73-95},
zbl = {1226.46052},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm204-1-5}
}
Ehrhard Behrends. Products of n open subsets in the space of continuous functions on [0,1]. Studia Mathematica, Tome 204 (2011) pp. 73-95. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm204-1-5/