On the Tychonoff product property of hereditarily ultraparacompact space

This paper obtains the following conclusions: (1) let X = Πσ ∈ΣXσbe hereditarily | Σ |- ultraparacompact space, then X is hereditarily ultraparacompact space if Πσ ∈ FXσ is hereditarily ultraparacompact space for every F ∈ [ Σ ]ω; (2) Let X =Πσ ω Xσ be hereditarily coun-table ultraparacompact, then the followings are equivalent: X is hereditarily ultraparacompact space; F ∈ [ω ]ω, Πi ∈ F Xi is hereditarily ultraparacompact space; Πi ≤n Xi is hereditarily ultrapar-acompact space for each n ∈ ω.