单点可度量化粗化：度量与局部度量保持
One-Point Metrizable Coarsenings: Gauges and Local Metric Preservation
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- Tarbiat Modares University（塔比阿特莫达雷斯大学）
- Institute for Research in Fundamental Sciences (IPM)（基础科学研究所）
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中文总结 AI 辅助
该文构造性地描述了在去掉一点后与原度量拓扑一致的可度量化粗化拓扑，利用Hausdorff度量塌缩得到满足指定边界的最优度量，并通过连续标量度量表示所有此类粗化。
中文摘要 AI 辅助
设 $(X,\tau)$ 为可度量化空间，且 $a\in X$。我们给出可度量化拓扑 $\sigma\subseteq\tau$ 的构造性描述，这些拓扑在 $X\setminus\{a\}$ 上与 $\tau$ 一致。应用豪斯多夫的经典度量塌缩构造，对于每个非紧空间 $(X,\tau)$ 和每个相容度量 $d$，我们得到一个严格粗化，其度量 $p\le d$ 在除 $a$ 外每一点的某个公共邻域上与 $d$ 一致。一个预先给定的可数无穷闭离散集 $\{x_n:n\in\N\}\subseteq X\setminus\{a\}$ 可以被使得满足 $p(a,x_n)\le\lambda_n$，其中 $(\lambda_n)$ 为任意正零序列。所得度量是在被 $d$ 控制且满足这些界限的度量中最大的，并且当 $d$ 完备时它也是完备的。我们展示了它作为经典度量商的具体实现。我们还利用标准锥度量，通过连续标量度量来表示所有局部化的可度量化粗化。包含关系通过扩展-迹理论中常见的子水平集的共尾比较来表达，而逐点最大和最小实现有限并和交。一个闭离散准则检测严格性。关于Borel结构、完全可度量化性和波兰空间的的标准保持结果，连同函数空间和局部域的例子，完成了这一描述。
英文摘要
Let $(X,τ)$ be metrizable and let $a\in X$. We give a constructive account of metrizable topologies $σ\subseteqτ$ that agree with $τ$ on $X\setminus\{a\}$. Applying Hausdorff's classical metric collapse construction, for every noncompact $(X,τ)$ and every compatible metric $d$ we obtain a strict coarsening with a metric $p\le d$ that agrees with $d$ on a common neighborhood of each point other than $a$. A prescribed countably infinite closed discrete set $\{x_n:n\in\N\}\subseteq X\setminus\{a\}$ can be made to satisfy $p(a,x_n)\leλ_n$ for any positive null sequence $(λ_n)$. The resulting metric is greatest among the metrics dominated by $d$ that satisfy these bounds, and is complete whenever $d$ is complete. We exhibit its realization as a classical metric quotient. We also represent all localized metrizable coarsenings by continuous scalar gauges using a standard cone metric. Inclusion is expressed by the cofinal comparison of sublevel sets familiar from extension-trace theory, while pointwise maximum and minimum realize finite joins and meets. A closed-discrete criterion detects strictness. Standard preservation results for Borel structure, complete metrizability, and Polishness, together with function-space and local-field examples, complete the account.
