Regular closed maps

A subset $A$ of $X$ is far from the remainder if whenever $\mathcal{U}$ is a free open ultrafilter in $X$ there exists $U \in \mathcal{U}$ such that $A \cap {\text{c}}{{\text{l}}_X}U = \emptyset$. A map is regular closed provided that the image of every regular closed set is closed. In this note we use some recent results of G. Viglino to show that every map can be extended to a regular closed map with far from the remainder point inverses. We also relate these maps to several other interesting classes of maps.