发表机构
N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences; Department of General Topology and Geometry, Mechanics and Mathematics Faculty, M. V. Lomonosov Moscow State University（俄罗斯科学院乌拉尔分校 N.N.克拉索夫斯基数学力学研究所; 莫斯科国立大学机械数学系一般拓扑与几何教研室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究齐次线性有序空间的紧子集性质，证明特征与基数上界并构造达到界的例子，同时证明GO半拓扑群的可度量性或P-空间性质。
AI 中文摘要
齐次广义有序（GO）空间的每个紧子集的特征至多为$\omega_1$，基数至多为$2^{\omega_1}$；若这样的子集具有不可数特征，则整个空间的特征等于$\omega_1$，且其$\pi$-特征是可数的。我们构造了一个齐次的$\sigma$-紧线性有序空间（LOTS）$\mathbf{H}$，其中包含一个基数$2^{\omega_1}$的紧子集$\mathbf{S}$，该子集在每一点的特征为$\omega_1$，且其权重和Souslin数均为$2^{\omega_1}$；因此所得到的两个界都是精确的。我们证明了作为GO空间的半拓扑群是遗传仿紧的；此外，若它不是$P$-空间，则它是可亚度量的，具有可数特征，且其紧子集是可度量的。每个线性有序半拓扑群（更一般地，每个GO半拓扑群）要么是可度量的，要么是$P$-空间；对拓扑群同样成立。我们还证明了在序齐次LOTS中，每个紧子集都是第一可数的。
英文摘要
Every compact subset of a homogeneous generalized ordered (GO) space has character at most $ω_1$ and cardinality at most $2^{ω_1}$; if such a subset has uncountable character, then the character of the whole space equals $ω_1$ and its $π$-character is countable. We construct a homogeneous $σ$-compact linearly ordered space (LOTS) $\mathbf{H}$ containing a compact subset $\mathbf{S}$ of cardinality $2^{ω_1}$ whose character is $ω_1$ at every point and whose weight and Souslin number are both $2^{ω_1}$; thus both bounds obtained are sharp. We prove that a semitopological group that is a GO space is hereditarily paracompact; if, in addition, it is not a $P$-space, then it is submetrizable, has countable character, and its compact subsets are metrizable. Every linearly ordered semitopological group (and, more generally, every GO semitopological group) is either metrizable or is a $P$-space; the same holds for topological groups. We also show that in an order-homogeneous LOTS every compact subset is first countable.
