Nearly Weak Refinable Spaces

The following results are proved(1)A space X is nearly weak  refinable iff X is nearly discrately weak  refinable expandable and for every open cover U={Uα:α∈Λ} of X then there is a dense set DX and a V=∪n∈ωVn of open refinements of U such that for each x∈D there are n∈ω and α∈Λ with x∈Uα and st(x,Vn)∪βα;(2)let X=∏α∈λXα be |Σ|-paracompact,then it is weak  refinable space if ∏α∈FXα is weak  refinable space for every F∈[ω]ω;(3)and for countable paracompace X=∏α∈ΛXi,the followings are equivalent:X is weak  refinable;F∈[ω]ω,∏α∈FXi is nearly weak  refinable;n∈ω,∏inXi is weak  refinable space.