Some Properties Of Bounded Sober Spaces And Bounded Well-filtered Spaces

In this paper,some basic properties of bounded sober space,bounded well-filtered space and bounded d-space are discussed.For a T0 space X,it is proved that the following three conditions are equivalent:(ⅰ)The Smyth power space PS(X)of X is a bounded sober space;(ⅱ)For each irreducible subset which has an upper boundary A ∈ Irr(PS(X)),U ∈O(X),if ∩A(?)U,then there is K ∈ A such that K(?)U;(ⅲ)For each irreducible subset A ∈ Irr(PS(X)),U ∈O(X),if ∩A ≠0 and ∩A(?)U,then there is K∈A such that K C U.For a To space X,it is proved that if PS(X)is a bounded sober space,then X is a bounded sober space;An example is given to show that there is a bounded sober space X such that its Smyth power space PS(X)is not a bounded well-filtered space,and hence PS(X)is not a bounded sober space.Also an example is given to show that the bounded sober space is not a bounded well-filtered space in general.For a T0 space X,it is proved that X is a bounded well-filtered space iff PS(X)is a bounded well-filtered space iff PS(X)is a bounded d-space.For a family {Xi:i∈I} of T0spaces,it is shown that the product space(?)Xi is a bounded d-space iff each factor space Xi is a bounded d-space.