The Weak Form of Normality / A. P. Kombarov. // Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika. 2017. № 5. P. 48-51
[Moscow Univ. Math. Bulletin. Vol. 72, N 2, 2017. P. 203-205].
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 \cal{F}:\text{Comp}\to \text{Comp} is a normal
functor of degree \geq 3 and the compact space {\cal{F}}(X) is hereditarily paranormal,
then the compact space X is metrizable.
Key words:
normal functor, compact space, hereditarily paranormality, metrizability.
