Geodesic
Topologies of uniform convergence. The property in the sense of Arens--Dugundji and the sequential property
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 9 (2013), pp. 45-58

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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 $C(X,Y)$.
Keywords: mapping space, topology of uniform convergence, admissible topology, proper topology, sequentially proper topology. 
@article{IVM_2013_9_a5,
     author = {V. L. Timokhovich and D. S. Frolova},
     title = {Topologies of uniform convergence. {The} property in the sense of {Arens--Dugundji} and the sequential property},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {45--58},
     publisher = {mathdoc},
     number = {9},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2013_9_a5/}
}
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V. L. Timokhovich; D. S. Frolova. Topologies of uniform convergence. The property in the sense of Arens--Dugundji and the sequential property. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 9 (2013), pp. 45-58. http://geodesic.mathdoc.fr/item/IVM_2013_9_a5/