有界sober空间和有界well-filtered空间的若干性质

Some Properties of Bounded Sober Spaces and Bounded Well-filtered Spaces

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T_(0)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T_(0)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。

In this paper,some basic properties of bounded sober space,bounded well-filtered space and bounded d-space are discussed.For a T_(0)space X,it is proved that the following three conditions are equivalent:(1)The Smyth power space P_(S)(X)of Xis 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∈Asuch 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∈Asuch that K■U.For a T0space X,it is shown that if P_(S)(X)is a bounded sober space,then Xis a bounded sober space.Two examples are given to show that there are bounded sober spaces Xsuch that the spaces Xare not bounded well-filtered and the Smyth power spaces P_(S)(X)are not bounded sober.For a T_(0)space X,it is proved that the following three conditions are equivalent:(1)Xis a bounded well-filtered space;(2)The Smyth power space P_(S)(X)of Xis a bounded d-space;(3)P_(S)(X)is a bounded well-filtered space.