Nearly Weak (?) Refinable Spaces And Base Metacompact Spaces

In this thesis, the covering and mapping methods are applied to makepreliminary research of the nearly weak (?) refinable space,base-metacompact,hereditarily hyperparacompact spaces, and we obtain the following results:(1) A space X is nearly weak (?) refinable space iff X is nearly discretely weak(?) refinable expandable and for every open cover u={U_α:α∈Λ} of X, then thereis a dense set D(?)X and a V=U_n∈ωV_n of open refinements of U such that foreach x∈D there are n∈ωandα∈Λwith x∈U_αand st(x, V_n)(?)∪_{β≤α} U_β;(2) Let X=П_{σ∈∑} X_σbe |∑|-paracompact, then it is weak (?) refinable spaceifП_σ∈F X_σis weak (?) refinable space for every F∈[ω]^＜ω;(3) For countable paracompace X=П_i∈ω X_i, the followings are equivalent: Xis weak (?) refinable; (?)F∈[ω]^＜ω,П_i∈FX_i is nearly weak (?) refinable;(?)n∈ω,П_i≤n X_i is weak (?) refinable space;(4) Let f:X→Y be perfect mapping, if Y is base-metaparacompace, thenX is base-metaparacompace;(5) A space X is base-metacompact if and only if there is base B of X, with|B|=ω(X), such that every directed open cover of X has a point-finite refinementbase B', B'(?)B. such that x∈Int(st(x, B')) for any x∈X;(6) Obtain a group of equivalent characterizations of hereditarilyhyperparacompact spaces, we obtain a Tychonoff product theorems by the one ofcharacterizations.