Topologies of uniform convergence. The property in the sense of Arens–Dugundji and the sequential property

This contribution investigates the properties of the topologies $\tau_\mathrm{sup}$ and $\tau_\mathrm{inf}$, which are, respectively, the supremum and the infimum of the family of all topologies of uniform convergence defined on the set $C(X,Y)$ of continuous maps into metrizable space $Y$. The main result of the research are necessary and sufficient conditions for properness and admissibility in the terms of Arens-Dugundji obtained for the topology $\tau_\mathrm{inf}$. The article introduces the notion of sequentially proper topology and establishes necessary and sufficient conditions for sequential properness of the topology $\tau_\mathrm{inf}$. It also considers a special case when the greatest proper topology and the greatest sequentially proper topology coincide on the set $C(X,Y)$.