Abstract
We study properties of topologies τ sup and τ inf, which are, respectively, the supremum and infimum of the family of all topologies of uniform convergence defined on the set C(X, Y) of continuous maps into a metrizable space Y. The main result of the research consists in obtaining necessary and sufficient conditions for the properness and admissibility in the sense of Arens-Dugundji for the topology τ inf. In this paper, we introduce the notion of a sequentially proper topology and establish necessary and sufficient conditions for the sequential properness of the topology τ inf. We also consider a special case when the maximal proper topology and the maximal sequentially proper topology coincide on the set C(X, Y).
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Original Russian Text © V.L. Timokhovich and D.S. Frolova, 2013, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2013, No. 9, pp. 45–58.
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Timokhovich, V.L., Frolova, D.S. Topologies of uniform convergence. The property in the sense of Arens-Dugundji and the sequential property. Russ Math. 57, 37–48 (2013). https://doi.org/10.3103/S1066369X13090065
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DOI: https://doi.org/10.3103/S1066369X13090065