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• # Artykuł - szczegóły

## Studia Mathematica

2011 | 204 | 1 | 73-95

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

EN

### Abstrakty

EN
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 $B_{ε}(f₁) ⋯ B_{ε}(fₙ)$ for all ε > 0 ? ($B_{ε}$ 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

73-95

wydano
2011

### Twórcy

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