论文标题
副学组的完成较弱
Weak completions of paratopological groups
论文作者
论文摘要
给定一个$ T_0 $副学组$ g $和$ \ $ \数学c $连续的副学组的同构，我们定义了$ \ MATHCAL C $ - $ $ $ nemiclestion $ $ $ $ $ $ \ nathcal c [g）$ $ $ \ $ \ $ \ $ \ $ \ $ \ $ $ $ $ \ $ \ $ \ $ \ $ g $ g $ g g]满足$ T_0 $分离公理并具有某些通用属性。对于特殊类$ \ Mathcal c $，我们在$ g $上介绍了一些必要的条件，以使（SEMI）完成$ \ MATHCAL C [g）$和$ \ MATHCAL C [G] $ Be Hausdorff。 Also, we give an example of a Hausdorff paratopological abelian group $G$ whose $\mathcal C$-semicompletion $\mathcal C[G)$ fails to be a $T_1$-space, where $\mathcal C$ is the class of continuous homomorphisms of sequentially compact topological groups to paratopological groups.特别是，组$ g $包含$ω$结合的顺序紧凑型亚组$ h $，因此$ h $是一个拓扑组，但$ g $中的关闭$ g $并未成为子组。
Given a $T_0$ paratopological group $G$ and a class $\mathcal C$ of continuous homomorphisms of paratopological groups, we define the $\mathcal C$-$semicompletion$ $\mathcal C[G)$ and $\mathcal C$-$completion$ $\mathcal C[G]$ of the group $G$ that contain $G$ as a dense subgroup, satisfy the $T_0$-separation axiom and have certain universality properties. For special classes $\mathcal C$, we present some necessary and sufficient conditions on $G$ in order that the (semi)completions $\mathcal C[G)$ and $\mathcal C[G]$ be Hausdorff. Also, we give an example of a Hausdorff paratopological abelian group $G$ whose $\mathcal C$-semicompletion $\mathcal C[G)$ fails to be a $T_1$-space, where $\mathcal C$ is the class of continuous homomorphisms of sequentially compact topological groups to paratopological groups. In particular, the group $G$ contains an $ω$-bounded sequentially compact subgroup $H$ such that $H$ is a topological group but its closure in $G$ fails to be a subgroup.
