【摘要】 本文主要讨论有界sober空间、有界well-filtered空间和有界d-空间的基本性质，证明了对T₀空间X,以下三个条件等价：（1）X的Smyth幂空间P_S（X）是有界sober空间；（2）对任意有上界的既约子集A∈Irr（P_S（X））,U∈O（X）,若∩A■U,则存在K∈A使得K■U;（3）对任意既约子集A∈Irr（P_S（X））,U∈O（X）,若∩A≠■,且∩A■U,则存在K∈A使得K■U。证明了：若T₀空间X的Smyth幂空间P_S（X）是有界sober空间，则X是有界sober空间。给出了两个例子说明：与sober性不同，存在有界sober空间X,X不是有界well-filtered的，其Smyth幂空间P_S（X）不是有界sober空间。对T₀空间X,证明了以下三个条件等价：（1）X为有界well-filtered空间；（2）X的Smyth幂空间P_S（X）为有界d-空间；（3）P_S（X）为有界well-filtered空间。更多还原

【Abstract】 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:（1）The Smyth power space P_S（X） of X is a bounded sober space;（2）For each irreducible subset which has an upper boundary A∈Irr（P_S（X））, U∈O（X）, if ∩A■U, then there is K∈A such that K■U;（3）For each irreducible subset A∈Irr（P_S（X））, U∈O（X）, if ∩A≠■ and ∩A■U, then there is K∈A such that K■U. For a T₀ space X, it is shown that if P_S（X） is a bounded sober space, then X is a bounded sober space. Two examples are given to show that there are bounded sober spaces X such that the spaces X are not bounded well-filtered and the Smyth power spaces P_S（X） are not bounded sober. For a T₀ space X, it is proved that the following three conditions are equivalent:（1）X is a bounded well-filtered space;（2）The Smyth power space P_S（X） of X is a bounded d-space;（3）P_S（X） is a bounded well-filtered space.更多还原