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arXiv 1507.04413math.GNmath.GR
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局部$σ$ -紧可求长空间

Locally $σ$-compact rectifiable spaces

Fucai Lin, Jing Zhang, Kexiu Zhang

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摘要：

一个拓扑空间$G$称为{ \it可求长空间}，只要有一个满射同胚$φ:G\times G\rightarrow G\times G$和一个元素$e\in G$使得$π_{1}\circ φ=π_{1}$和对于每个$x\in G$，$φ(x, x)=(x, e)$，其中$π_{1}: G\times G\rightarrow G$是到第一个坐标的投影。本文首先证明了每个局部紧可求长空间都是仿紧的，从而肯定地回答了Arhangel'skii和Choban的问题（Arhangel'skii和Choban [3]）。然后证明了每一个具有$bc$ -基的局部$σ$ -紧可求长空间都是局部紧的或零维的，改进了Arhangel'skii和van Mill的结果（Arhangel'skii和van Mill [4]）.最后证明了每个$k_ω$ -可求长空间都是可求长完备的。

英文摘要：

A topological space $G$ is said to be a {\it rectifiable space} provided that there are a surjective homeomorphism $φ:G\times G\rightarrow G\times G$ and an element $e\in G$ such that $π_{1}\circ φ=π_{1}$ and for every $x\in G$, $φ(x, x)=(x, e)$, where $π_{1}: G\times G\rightarrow G$ is the projection to the first coordinate. In this paper, we first prove that each locally compact rectifiable space is paracompact, which gives an affirmative answer to Arhangel'skii and Choban's question (Arhangel'skii and Choban [3]). Then we prove that every locally $σ$-compact rectifiable space with a $bc$-base is locally compact or zero-dimensional, which improves Arhangel'skii and van Mill's result (Arhangel'skii and van Mill [4]). Finally, we prove that each $k_ω$-rectifiable space is rectifiable complete.

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