Of �� Refinable Space The Promotion And Weakly [�� <sub> 1 </ Sub>, ��] <sup>, R </ Sup> A Refinable The Nature,

In the paper, we give definitions of F_σweakθ-refinable spaces and pointwiseclosure-preserving spaces and these spaces are generalizations ofθ-refinable spaces. Asapplications of these spaces, we give necessary and su?cient conditions for collection-wise normal spaces to be paracompact spaces and for spaces which having G_δ-diagonalto have G_δ^*-diagonal respectively. The latter one generalizes a theorem of Kullman. Wealso discuss the properties of weakly [ω₁,∞)^γ-refinable spaces.