The aim of this paper is to study certain closure-type properties of function spaces over metric spaces endowed with two topologies: the topology of uniform convergence on a bornology and the topology of strong uniform convergence on a bornology. The study of function spaces with the strong uniform topology on a bornology was initiated by G. Beer and S. Levi in 2009, and then continued by several authors: A. Caserta, G. Di Maio and L'. Hola in 2010, A. Caserta, G. Di Maio, Lj. D. R. Kocinac in 2012. Properties that we consider in this paper are defined in terms of selection principles.