Normal functors and paranormality
It is proved that if ${\cal F}:{\cal P}\longrightarrow {\cal P}$ is a normal functor of degree $\geqslant 3$ in the category ${\cal P}$ of paracompact $p$-spaces and perfect mappings, and the space ${\cal F}(X)$ is hereditarily paranormal, then the space $X$ is metrizable.