The weak form of normality

A topological space is said to be paranormal if every countable discrete collection of closed sets {D _n : n < ω} can be expanded to a locally finite collection of open sets {U _n : n < ω}, i.e., D _n ? U _n and D _m ∩ U _n ≠ 0 if and only if D _m = D _n . It is proved that if F: Comp → Comp is a normal functor of degree ≥ 3 and the compact space F(X) is hereditarily paranormal, then the compact space X is metrizable.