Abstract In this paper, we introduce paratopogical left-loops, as natural generalizations of paratopogical left Bol loops defined by Z. Cai, S. Lin and W. He in [7] , and prove that some known results for paratopological groups are still valid for paratopogical left-loops. The main results are: (i) every weakly first-countable paratopological left-loop is first-countable; (ii) a sequential, csf-countable, regular paratopological left-loop is first-countable if and only if it contains no closed copy of S ω .

中文翻译：

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副拓扑左环的一阶可数性

摘要 在本文中，我们引入了平行拓扑左环，作为由 Z. Cai、S. Lin 和 W. He 在 [7] 中定义的平行拓扑左 Bol 环的自然推广，并证明了一些已知的平行拓扑群结果仍然有效用于平行拓扑的左环。主要结果是：（i）每个弱第一可数的并拓扑左环都是第一可数的；(ii) 一个连续的、csf 可数的、规则的并拓扑左环是第一可数的当且仅当它不包含 S ω 的封闭副本。