Axioms of Continuous Separation

Zhongqiang Yang; Nada Wu

paper

Axioms of Continuous Separation

arXiv:2608.13086

Abstract

For a $T_{1}$ -space $X$ , let $C l d (X)$ denote all its nonempty closed subsets and $T_{4} (X) = {(F_{1}, F_{2}) \in C l d (X)^{2} : F_{1} \cap F_{2} = \emptyset}$ . 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 $C l d (X)$ , we introduce the definition: A $T_{1}$ -space $X$ is called $C T_{4}$ for the topology $τ$ if the above maps $ϕ_{j} : T_{4} (X) \to C l d (X)$ are continuous on $τ$ . Similarly, for $i = 1, 2, 3$ , we can define a $T_{1}$ -space to be $C T_{i}$ for $τ$ . We only consider the Vietoris topology on $C l d (X)$ and show that every $C T_{4}$ -space is countably compact, and every $C T_{3}$ -space is a Fréchet-Urysohn space, every separable subspace of a $C T_{3}$ -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 $C T_{4}$ . All infinite discrete spaces, all finite-dimensional Euclidean spaces and all countable limit ordinal spaces are $C T_{3}$ but not $C T_{4}$ . Also, each metrizable space with a unique non-isolated point is $C T_{3}$ , and is $C T_{4}$ if it is compact. Moreover, the infinite sum of $C T_{3}$ spaces is $C T_{3}$ but not $C T_{4}$ . Every countable space with a unique non-isolated point is $C T_{2}$ , and it is $C T_{3}$ if and only if it is metrizable. All subfields of real numbers and their complement spaces are $C T_{2}$ but not $C T_{4}$ ; and their $C T_{3}$ status remains unclear. All uncountable ordinal spaces are not $C T_{2}$ . The one-point compactification of any uncountable discrete space is not $C T_{2}$ . $C T_{1}$ and $T_{1}$ are equivalent, hence all spaces above are $C T_{1}$ . Open Problems: is there a non-metrizable $C T_{3}$ or $C T_{4}$ space? Is any compact $C T_{2}$ space $C T_{3}$ or $C T_{4}$ ?