$\tau$-pseudocompact mappings

We consider the problem of extending the notion of $\tau$-pseudocompactness from spaces to continuous mappings, obtain conditions under which the product of $\tau$-pseudocompact mappings is $\tau$-pseudocompact. Since any space $X$ can be considered as a continuous mapping from $X$ into a singleton, we obtain consequences of the theorems on multiplicativity of $\tau$-pseudocompactness for spaces. Thus, we study the notion of $\tau$-pseudocompact mapping and some its properties similar to those of a seudocompact space as well as consequences of the obtained assertions for spaces.