When does argmin multifunction have a closed graph?

Abstract Let X be a Tychonoff topological space, C (X) be the space of continuous real-valued functions defined on X and K (X) be the space of all nonempty compact subsets of X. Define the multifunction argmin: C (X) × K (X) → X as follows: argmin (f, K) = x ∈ K: f (x) = min{ f (y): y ∈ K }. Using bornologies with closed bases we present interesting topologies on C (X) × K (X) under which argmin: C (X) × K (X) → X has a closed graph and the set (f, K): |argmin (f, K) | = 1 is dense in C (X) × K (X). On the function space C (X) we consider the topology of uniform convergence τ B ₁ on a bornology B B and on the space K (X) we consider hit-and-miss hyperspace topology generated by a closed base of B B.