Minimal cusco maps have applications in functional analysis, in optimization, in the study of weak Asplund spaces, in the study of differentiability of functions, etc. It is important to know their topological properties. Let X be a Hausdorff topological space, MC(X) be the space of minimal cusco maps with values in R and τ UC be the topology of uniform convergence on compacta. We study complete metrizability and cardinal invariants of (MC(X), τ UC). We prove that for two nondiscrete locally compact second countable spaces X and Y, (MC(X), τ UC) and (MC(Y), τ UC) are homeomorphic and they are homeomorphic to the space C(I c) of continuous real-valued functions on I c with the topology of uniform convergence.