The weak form of normality

A. P. Kombarov

Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (2017), pp. 48-51

Résumé

A topological space is said to be paranormal if every countable discrete collection of closed sets $\{D_n: n\omega\}$ can be expanded to a locally finite collection of open sets $\{U_n: n\omega\}$, i.e., $D_n\subset U_n$ and $D_m\cap U_n\not=\emptyset$ if and only if $D_m=D_n$. It is proved that if $\mathcal{F}:$ Comp $ \to$ Comp is a normal functor of degree $\geq 3$ and the compact space ${\mathcal{F}}(X)$ is hereditarily paranormal, then the compact space $X$ is metrizable.