发表机构
Guangxi Minzu University; Hanshan Normal University（广西民族大学; 韩山师范学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对Vietoris拓扑定义了CT_i空间，证明了CT₄空间可数紧、CT₃空间是Fréchet-Urysohn空间等性质，给出多类空间的CT_i判定结果，并提出两个开放问题。
AI 中文摘要
对于T₁空间X，令Cld(X)表示其所有非空闭子集，T₄(X)={(F₁,F₂)∈Cld(X)²:F₁∩F₂=∅}。就闭集而言，X是T₄空间当且仅当对每一组(F₁,F₂)∈T₄(X)，存在一对闭集φ₁(F₁,F₂)和φ₂(F₁,F₂)，使得它们的并集为X，且对j=1,2，有F_j∩φ_j(F₁,F₂)=∅。因此，本文针对Cld(X)上的拓扑τ，引入如下定义：对于拓扑τ，T₁空间X被称为τ下的CT₄空间，当上述映射φ_j:T₄(X)→Cld(X)在τ上连续。类似地，对于i=1,2,3，可定义τ下的CT_i空间。本文仅考虑Cld(X)上的Vietoris拓扑，证明了每个CT₄空间是可数紧的，每个CT₃空间是Fréchet-Urysohn空间，CT₃空间的每个可分子空间都是可度量化的；并给出了相关例子：所有有限维立方体、无限维立方体、所有有限维球面以及所有0维紧可度量化空间都是CT₄空间；所有无限离散空间、所有有限维欧氏空间和所有可数极限序数空间都是CT₃空间但非CT₄空间；每个具有唯一非孤立点的可度量化空间是CT₃空间，且若该空间紧则为CT₄空间；CT₃空间的无限和是CT₃空间但非CT₄空间；每个具有唯一非孤立点的可数空间是CT₂空间，且当且仅当它可度量化时为CT₃空间；实数的所有子域及其补空间都是CT₂空间但非CT₄空间，其CT₃状态仍不明确；所有不可数序数空间都不是CT₂空间；任意不可数离散空间的一点紧化不是CT₂空间；CT₁与T₁等价，因此上述所有空间都是CT₁空间。开放问题：是否存在非可度量化的CT₃或CT₄空间？任意紧CT₂空间是CT₃或CT₄空间吗？
英文摘要
For a $T_1$-space $X$, let $Cld(X)$ denote all its nonempty closed subsets and $T_4(X)=\{(F_1,F_2)\in Cld(X)^2:F_1\cap F_2=\emptyset\}$. In terms of closed sets, $X$ is $T_4$ iff for each $(F_1,F_2)\in T_4(X)$, there is a pair of closed sets $ϕ_1(F_1,F_2)$ and $ϕ_2(F_1,F_2)$ such that their union is $X$ and $F_j\cap ϕ_j(F_1,F_2)=\emptyset$ for $j=1,2$. Thus, in this paper, for a topology $τ$ on $Cld(X)$, we introduce the definition: A $T_1$-space $X$ is called $CT_4$ with respect to $τ$ if the above maps $ϕ_j:T_4(X)\to Cld(X)$ are continuous on $τ$. Similarly, for $i=1,2,3$, we can define a $T_1$-space to be $CT_i$ for $τ$. We only consider the Vietoris topology on $Cld(X)$ and show that every $CT_4$-space is countably compact, and every $CT_3$-space is a Fréchet-Urysohn space, every separable subspace of a $CT_3$-space is metrizable. We give relevant examples. Any finite-dimensional cubes, the infinite-dimensional cube, any finite-dimensional spheres, and all 0-dimensional compact metrizable spaces are $CT_4$. All infinite discrete spaces, all finite-dimensional Euclidean spaces and all countable limit ordinal spaces are $CT_3$ but not $CT_4$. Also, each metrizable space with a unique non-isolated point is $CT_3$, and is $CT_4$ if it is compact. Moreover, the infinite sum of $CT_3$ spaces is $CT_3$ but not $CT_4$. Every countable space with a unique non-isolated point is $CT_2$, and it is $CT_3$ if and only if it is metrizable. All subfields of real numbers and their complement spaces are $CT_2$ but not $CT_4$; and their $CT_3$ status remains unclear. All uncountable ordinal spaces are not $CT_2$. The one-point compactification of any uncountable discrete space is not $CT_2$. $CT_1$ and $T_1$ are equivalent, hence all spaces above are $CT_1$. Open Problems: is there a non-metrizable $CT_3$ or $CT_4$ space? Is any compact $CT_2$ space $CT_3$ or $CT_4$?
