| 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.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 208, 1993, pp. 82–97.
Translated by A. A. Ivanov. |
