A study concerning splitting and jointly continuous topologies onC(Y,Z)

Abstract

Let Y and Z be two fixed topological spaces and C(Y, Z) the set of all continuous maps from Y into Z. We construct and study topologies on C(Y, Z) that we call F n (τ n )-family-open topologies. Furthermore, we find necessary and sufficient conditions such that these topologies to be splitting and jointly continuous. Finally, we present questions concerning a further study on this area.

Robots That Do Not Avoid Obstacles

“…Here we ask for the possibility of investigating purely topological properties of robot motion planning algorithms via function spaces, based on the study in [4] and on the results by Farber. Considering a function space F(X, Y ), there are several topological problems one can study.…”

“…Start with a topological space X, as the configuration space of a mechanical system, with no explicit information about its local or global topological properties. Apply Step 0 to Step n of the construction given in [4], to the motion planning algorithms space F(X × X, P X). Study the possibility for the existence of a minimal integer n "revealing as much as possible topological information about X".…”