Fixed points and best approximation in arcwise connected spaces

It is shown that several theorems known to hold in complete geodesically bounded RR-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][u,v] denote the unique arc joining u,v∈Xu,v∈X. Among other things, it is shown and if Y is a closed connected subset of X   and if f:Y→Xf:Y→X is continuous, then f has a ‘best approximation’ in Y   in the sense that there exists a point z∈Yz∈Y such that [z,f(z)]∩Y={z}[z,f(z)]∩Y={z}. A set-valued analog of this result is also discussed.