Abstract
It is shown that several theorems known to hold in complete geodesically bounded R-trees extend to arcwise connected Hausdorff topological spaces which have the property that every monotone increasing sequence of arcs is contained in an arc. Let X be such a space and let [u,v] denote the unique arc joining u,v∈X. Among other things, it is shown and if Y is a closed connected subset of X and if f:Y→X is continuous, then f has a ‘best approximation’ in Y in the sense that there exists a point z∈Y such that [z,f(z)]∩Y={z}. A set-valued analog of this result is also discussed.
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