The inverse limits proposition of normal weakly θ-refinable spaces

This paper proves the following results:Let X= lim ←{X σ,π σ ρ,∧},｜∧｜=λ,and every π σ be open and onto mapping,(1) if X is λparacompact and every X σ is normal and weakly θrefinable,then X is normal and weakly θrefinable;(2) if X is hereditarily λ paracompact and every X σ is hereditarily normal and hereditarily weakly θrefinable,then X is hereditarily normal and hereditarily weakly θrefinable.