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连通和序列紧的 Rectifiable 空间
Connected and Sequentially Compact Rectifiable Spaces

张静,贺伟
ZHANG Jing*,HE Wei

南京师范大学数学科学学院, 南京, 江苏, 210046
Institute of Mathematics, Nanjing Normal University, Nanjing, Jiangsu, 210046, P. R. China

收稿日期: 2013-11-08
出版日期: 2015-07-25
DOI: 10.11845/sxjz.2013146b

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摘要 本文主要讨论了~rectifiable~空间的连通, 序列紧和 ~$\kappa$-Fr\'{e}chet-Urysohn性质. 证明了以下结果: (1) 若~$G$~是局部~$\sigma$-序列紧且具有~Souslin~性质的~rectifiable 空间, 则~$G$~是~$\sigma$-序列紧的. (2) 每一连通的局部~$\sigma$-紧的~rectifiable~ 空间~$G$~是~$\sigma$-紧的. (3) 若 ~rectifiable空间~$G$~ 的每一紧(可数紧, 序列紧)的子空间是~Fr\'{e}chet-Urysohn, 则~$G$ 的每一紧(可数紧,~序列紧)~的子空间是强~Fr\'{e}chet-Urysohn. 这些结果推广了拓扑群中的相应结果.
关键词 rectifiable~空间连通空间序列紧空间$\kappa$-Fr\'{e}chet-Urysohn空间    
Abstract:In this note, we mainly investigate the connected, sequentially compact and $\kappa$-Fr\'{e}chet-Urysohn properties of rectifiable spaces. It is showed that: (1) If $G$ is a locally $\sigma$-sequentially compact rectifiable space with the Souslin property, then $G$ is $\sigma$-sequentially compact; (2) Every connected locally $\sigma$-compact rectifiable space $G$ is $\sigma$-compact; (3) If every compact (resp. countably compact, sequentially compact) subspace of a rectifiable space $G$ is Fr\'{e}chet-Urysohn, then every compact (resp. countably compact, sequentially compact) subspace of $G$ is strongly Fr\'{e}chet-Urysohn. These results generalize the corresponding results in topological groups.
Key wordsconnected space    sequentially compact space    $\kappa$-Fr\'{e}chet-Urysohn space
[1] 朴勇杰. 聚合不动点定理及其对相交和变分不等式问题的应用[J]. 数学进展, 2014, 43(3): 435-444.
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