Abstract
In the present paper we introduce notions of A-splitting and A-jointly continuous topology on the set C(Y,Z) of all continuous maps of a topological space Y into a topological space Z, where A is any family of spaces. These notions satisfy the basic properties of splitting and jointly continuous topologies on C(Y,Z). In particular, for every A, the greatest A-splitting topology on C(Y,Z) (denoted by τ(A) always exists. We indicate some families A of spaces for which the topology τ(A) coincides with the greatest splitting topology on C(X,Y). We give a notion of equivalent families of spaces and try to find a “simple” family which is equivalent to a given family. In particular, we prove that every family is equivalent to a family consisting of one space, and the family of all spaces is equivalent to a family of all T1-spaces containing at most one nonisolated point. We compare the topologies τ({X}) for distinct compact metrizable spaces X and give some examples. Bibliography: 13 titles.
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Additional information
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 208, 1993, pp. 82–97.
Translated by A. A. Ivanov.
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Georgiou, D.N., Iliadis, S.D. & Papadopoulos, B.K. Topologies on function spaces. J Math Sci 81, 2506–2514 (1996). https://doi.org/10.1007/BF02362419
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DOI: https://doi.org/10.1007/BF02362419