We identify two key conditions that a subset $A$ of a poset $P$ may satisfy to guarantee the transfer of continuity properties from $P$ to $A$. We then highlight practical cases where these key conditions are fulfilled. Along the way we are led to consider subsets of a given poset $P$ whose way-below relation is the restriction of the way-below relation of $P$, which we call way-below preserving subposets. As an application, we show that every conditionally complete poset with the interpolation property contains a largest continuous way-below preserving subposet. Most of our results are expressed in the general setting of Z theory, where Z is a subset system.
翻译:我们确定了两个关键条件,即一个重质美元子A美元可以满足于保证连续性财产从P美元转移至A美元。然后我们强调了这些关键条件得到满足的实际案例。在引导我们考虑一个特定重质P美元子项目的方式上,其方式关系较低的是限制低价美元的关系,我们称之为“低价”保护子质。作为一个应用程序,我们证明每个有条件完整的内插财产都含有一个最大的连续持续方式,而保护子质以下。我们的结果大多体现在Z理论的总体背景中,Z是子项体系。