超度量空间模空间的绝对Borel复杂度
Absolute Borel Complexity of Moduli Spaces of Ultrametrics
AI总结：
研究可超度量化空间上有界相容超度量模空间的绝对Borel复杂度，证明其与空间X对应层级复杂度的等价性，得出该模空间完全可度量化的充要条件。
AI中文摘要：
设X为可超度量化空间，研究其上带自然非阿基米德距离的有界相容超度量构成的空间。对每个正整数层级，证明该模空间的加法绝对Borel复杂度蕴含X在同一层级的乘法绝对Borel复杂度，反之亦然。还证明可超度量化空间是局部紧子空间的可数并，当且仅当它是每个由有界相容超度量诱导的完备化中闭子集的可数并。由此得出，该模空间完全可度量化当且仅当X是紧子集的可数并。
英文摘要：
Let $X$ be an ultrametrizable space. We study the space of bounded compatible ultrametrics on $X$, equipped with its natural non-Archimedean distance. For every positive integer level, we prove that additive absolute Borel complexity of this moduli space implies multiplicative absolute Borel complexity of $X$ at the same level, and conversely. We also prove that an ultrametrizable space is a countable union of locally compact subspaces if and only if it is a countable union of closed subsets in every completion induced by a bounded compatible ultrametric. As a consequence, this moduli space is completely metrizable exactly when $X$ is a countable union of compact subsets.
