Fixed points and best approximation in arcwise connected spaces

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.