Bifactorial mapping

A mapping $f$ of a topological space $X$ into a topological space $Y$, in which for any covering of the inverse image $f^{-1}(y)$ of any point $y\in f(X)$ by sets open in $X$ it is possible to select a finite number of sets so that $y$ is located inside the image of their union. It is particularly important that the product of any collection of bifactorial mappings is a bifactorial mapping. Bifactorial mappings constitute an extensive class of factorial mappings, but nevertheless preserve the fine topological properties of spaces. Thus, continuous bifactorial $s$-mappings preserve a pointwise-countable base, and a factorial $s$-mapping of a space with a pointwise-countable base onto a space of pointwise-countable type is bifactorial.