| In this paper,using countably-directed sets,we introduce the concepts of ω*-Rudin spaces and ω*-well-filtered determined spaces,and discuss some basic prop-erties of ω*-well-filtered spaces and related spaces.Some characterizations of ω*-well-filteredness are given and the following results are obtained:(1)A retract of anω*-Rudin space is an ω*-Rudin space;(2)A retract of an ω*-well-filtered determined space is an ω*-well-filtered determined space;(3)For a finite family {Xi:1≤i≤n}of T0 spaces,the product space Πi=1n Xi is ω*-well-filtered determined space if and only if for any i=1,2,...n,Xi is ω*-well-filtered determined space;(4)The retract of an ω*-well-filtered space is an ω*-well-filtered space;(5)For a family {Xi:i ∈ I}of T0 spaces,the product space Πi∈I Xi is ω*-well-filtered space if and only if for any i ∈I,Xi is ω*-well-filtered space;(6)The category of all ω*-well-filtered space is reflective in that of all T0 spaces,and the concrete ω*-well-filtered reflective of a T0 space is given. |
