In the present paper we introduce the notions of $\mathcal A$-splitting and $\mathcal A$-jointly continuous topologies on the set $C(Y,Z)$ of all continuous maps of a topological space $Y$ into a topological space $Z$, where $\mathcal A$ 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 $\mathcal A$, the greatest $\mathcal A$-splitting topology on $C(Y,Z)$ (denoted by $\tau(\mathcal A)$) always exists. We indicate some families $\mathcal A$ of spaces, for which the topology $\tau(\mathcal A)$ coinsides with the greatest splitting topology on $C(Y,Z)$. We give the notion of equivalent families of spaces and try to define 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 $T_1$-spaces containing at most one non-isolated point. We compare the topologies $\tau(\{X\})$ for distinct compact metrizable spaces $X$ and give some examples. Bibliography: 13 titles.