链并闭包
Chain union closures
AI总结：
本文研究球空间球面完备性在链并闭包下的保持条件，给出适用于线性序或可数窄值集超度量空间的判据，并引入链并稳定性与链并秩来刻画闭包迭代次数。
摘要：
我们研究球空间的球面完备性及其在扩张下的稳定性。我们给出球空间的一些判据，以保证当球空间在链的并运算下封闭时，球面完备性得以保持。这尤其适用于具有线性序值集的超度量空间中的闭超度量球空间，或更一般地，适用于具有可数窄值集的超度量空间。我们证明，一般而言，具有偏序值集的超度量空间的链并闭包并不保持球面完备性。此外，我们引入并研究了链并稳定性与链并秩的概念，它们度量了将球空间在所有链的并运算下封闭的过程需要迭代多少次，才能得到一个在链的并运算下封闭的球空间。
英文摘要：
We study spherical completeness of ball spaces and its stability under expansions. We give some criteria for ball spaces that guarantee that spherical completeness is preserved when the ball space is closed under unions of chains. This applies in particular to the spaces of closed ultrametric balls in ultrametric spaces with linearly ordered value sets, or more generally, with countable narrow value sets. We show that in general, chain union closures of ultrametric spaces with partially ordered value sets do not preserve spherical completeness. Further, we introduce and study the notions of chain union stability and of chain union rank, which measure how often the process of closing a ball space under all unions of chains has to be iterated until a ball space is obtained that is closed under unions of chains.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
