On the relation between continuous functions in two different metric spaces

2014/06/11 by Adrian Fellhauer, Fellhauer, Adrian
Mathematics · #FOS: Mathematics #Metric Geometry (math.MG) #math.MG

paper · pdf · doi:10.48550/arxiv.1406.2805

4 pages; the result had not achieved full generality yet; small mistake on page 4; some corrections regarding to the english language as kindly suggested by a friend of mine

arxiv created 2014/06/23 · arxiv updated 2014/06/24

Abstract

Let the metric space \mathbb Rⁿ ∖ ∼ be the metric space of n-sized unordered tuples of real numbers. In the following, it will be shown that if a function φ: \mathbb R^m → \mathbb Rⁿ ∖ ∼ is continuous, then there is a continuous function f: \mathbb R^m → \mathbb Rⁿ such that a natural embedding of f into \mathbb Rⁿ ∖ ∼ is equal to φ. This theorem is wrong in the complex case. A counterexample is given in [1].

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