论文标题
封闭子集具有可计数特征的高空空间
Hyperspaces with a countable character of closed subsets
论文作者
论文摘要
对于常规空间$ x $，Hyperspace $（Cl（x），τ_{f}）$（分别，$（cl（x），τ_{v}）$）是所有非空的封闭子集的空间，其中$ x $，带有fell Toupology（resp。Vietoris，Vietoris Topology）。在本文中，我们给出了空间$ x $的表征，使得超空间$（cl（x），τ_{f}）$（resp。，$（cl（x），τ_{v}）$）具有可数封闭子集的特征。我们主要证明$（cl（x），τ_f）$在每个闭合子集上都有一个可数的字符，并且仅当$ x $是紧凑的MetrizAble，而$（cl（x），τ_f）$在每个紧凑型子集中具有可数值的字符时，并且仅当$ x $是局部紧凑和可分离的Metrizrize的情况时，每个紧凑型子集具有可数值的字符。此外，我们证明$（\ Mathcal {k}（x），τ_v）$具有紧凑型$g_δ$属性，并且仅当$ x $具有紧凑型$g_δ$属性，并且每一个紧凑型子集的$ x $的每个紧凑型子集均为metrizable。
For a regular space $X$, the hyperspace $(CL(X), τ_{F})$ (resp., $(CL(X), τ_{V})$) is the space of all nonempty closed subsets of $X$ with the Fell topology (resp., Vietoris topology). In this paper, we give the characterization of the space $X$ such that the hyperspace $(CL(X), τ_{F})$ (resp., $(CL(X), τ_{V})$) with a countable character of closed subsets. We mainly prove that $(CL(X), τ_F)$ has a countable character on each closed subset if and only if $X$ is compact metrizable, and $(CL(X), τ_F)$ has a countable character on each compact subset if and only if $X$ is locally compact and separable metrizable. Moreover, we prove that $(\mathcal{K}(X), τ_V)$ have the compact-$G_δ$ property if and only if $X$ have the compact-$G_δ$ property and every compact subset of $X$ is metrizable.
