Spaces and maps approximation and fixed points

A general approximation property for topological spaces is studied in relation with fixed point theory for set-valued maps. A particular instance of this property is the admissibility in the sense of Klee. Examples of “convex” sets of topological spaces equipped with a local topological convexity structure as well as general classes of approximative neighborhood retracts are shown to have this approximation property. A general topological principle on the preservation of the fixed point property under this space approximation is proved. It allows the passage from basic classes of spaces to more elaborate ones for general classes of nonconvex set-valued maps.