The principle that "any product of cofinite topologies is compact" is equivalent (without appealing to the Axiom of Choice) to the Boolean Prime Ideal Theorem.

This paper completes and improves results of [10]. Let $(X,d_{{}_X})$, $(Y,d_{{}_Y})$ be two metric spaces and $G$ be the space of all $Y$-valued continuous functions whose domain is a closed subset of $X$. If $X$ is a locally compact metric space, then the Kuratowski convergence ${\tau }_{{}_K}$ and the Kuratowski convergence on compacta ${\tau }_{{}_K}^c$ coincide on $G$. Thus if $X$ and $Y$ are boundedly compact metric spaces we have the equivalence of the convergence in the Attouch-Wets topology ${\tau }_{{}_{AW}}$ (generated by the box metric of $d_{{}_X}$ and $d_{{}_Y}$) and ${\tau }_{{}_K}^c$ convergence on $G$,...