发表机构
Tarbiat Modares University; Institute for Research in Fundamental Sciences (IPM)（塔比阿特·莫达雷斯大学; 伊朗基础科学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究可度量化空间在有限点集上粗化拓扑的刻画，通过相容度量系统给出显式度量，并证明交拓扑的相容性等价于共同Hausdorff下界，进而刻画了格结构仅存在于紧空间的情形。
AI 中文摘要
设$(X,\tau)$为可度量化空间，$F=\{a_1,\ldots,a_k\}\subseteq X$，其中$2\le k<\infty$。我们通过相容的连续度量系统$s_i:X\to[0,1]$（满足$s_i^{-1}(0)=\{a_i\}$）来刻画所有在$X\setminus F$上与$\tau$一致的可度量化拓扑$\sigma\subseteq\tau$。条件$\inf_X\max\{s_i,s_j\}>0$（对$i\ne j$）等价于所规定度量拓扑的Hausdorff性和可度量化性。归一化乘积映射到标准单纯形给出了显式度量；其三角不等式由单纯形松弛不等式得出。当辅助有界相容度量完备时，该度量完备。对于两个相容系统，其逐坐标最小值描述了交拓扑。当且仅当两个粗化具有共同的Hausdorff下界时，该交拓扑是相容的；此时交拓扑可度量化且为两者的交。否则，每个共同的下拓扑都是非Hausdorff的。闭离散构造为每个非紧可度量化空间和每个至少含两点的有限例外集产生这样的受阻对。因此，对于这些例外集，该族是向下有向的，或是格，当且仅当$(X,\tau)$是紧的，此时该族仅由$\tau$组成。
英文摘要
Let $(X,τ)$ be metrizable and let $F=\{a_1,\ldots,a_k\}\subseteq X$, where $2\le k<\infty$. We represent all metrizable topologies $σ\subseteqτ$ agreeing with $τ$ on $X\setminus F$ by compatible systems of continuous gauges $s_i:X\to[0,1]$ with $s_i^{-1}(0)=\{a_i\}$. The condition $\inf_X\max\{s_i,s_j\}>0$ for $i\ne j$ is equivalent to both Hausdorffness and metrizability of the prescribed gauge topology. A normalized product map into the standard simplex gives an explicit metric; its triangle inequality follows from a simplex slack inequality. This metric is complete whenever the auxiliary bounded compatible metric is complete. For two compatible systems, their coordinatewise minimum describes the intersection topology. It is compatible exactly when the two coarsenings have a common Hausdorff lower bound; in that case the intersection is metrizable and is their meet. Otherwise every common lower topology is non-Hausdorff. A closed-discrete construction produces such an obstructed pair for every noncompact metrizable space and every finite exceptional set with at least two points. Consequently, for these exceptional sets, the family is downward directed, or is a lattice, if and only if $(X,τ)$ is compact, in which case it consists only of $τ$.
Comments13 pages. Comments welcome
