Arzelà's theorem and strong uniform convergence on bornologies (Q986604)

The main result of this nice paper (Theorem 2.9) gives a direct proof that three kinds of convergence (Arzelà's convergence on compacta, Alexandroff's convergence and strong uniform convergence on finite sets, introduced recently by \textit{G. Beer} and \textit{S. Levi} [J. Math. Anal. Appl. 350, 568--589 (2009; Zbl 1161.54003)]) of nets of continuous functions between two metric spaces coincide, and that each of these conditions is equivalent to continuity of the limit function. Several properties ((sub)metrizability, the \(G_\delta\)-diagonal property, countable pseudocharacter, separability, second countability, the Fréchet-Urysohn property) of the topology of strong uniform convergence on a bornology are also established.