Axioms of Continuous Separation
Abstract
For a T1-space X, let Cld(X) denote all its nonempty closed subsets and T4(X)={(F1,F2)∈ Cld(X)2:F1∩ F2=∅}. In terms of closed sets, X is T4 iff for each (F1,F2)∈ T4(X), there is a pair of closed sets ϕ1(F1,F2) and ϕ2(F1,F2) such that their union is X and Fj∩ ϕj(F1,F2)=∅ for j=1,2. Thus, in this paper, for a topology τ on Cld(X), we introduce the definition: A T1-space X is called CT4 with respect to τ if the above maps ϕj:T4(X)→ Cld(X) are continuous on τ. Similarly, for i=1,2,3, we can define a T1-space to be CTi for τ. We only consider the Vietoris topology on Cld(X) and show that every CT4-space is countably compact, and every CT3-space is a Fréchet-Urysohn space, every separable subspace of a CT3-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 CT4. All infinite discrete spaces, all finite-dimensional Euclidean spaces and all countable limit ordinal spaces are CT3 but not CT4. Also, each metrizable space with a unique non-isolated point is CT3, and is CT4 if it is compact. Moreover, the infinite sum of CT3 spaces is CT3 but not CT4. Every countable space with a unique non-isolated point is CT2, and it is CT3 if and only if it is metrizable. All subfields of real numbers and their complement spaces are CT2 but not CT4; and their CT3 status remains unclear. All uncountable ordinal spaces are not CT2. The one-point compactification of any uncountable discrete space is not CT2. CT1 and T1 are equivalent, hence all spaces above are CT1. Open Problems: is there a non-metrizable CT3 or CT4 space? Is any compact CT2 space CT3 or CT4?
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BibTeX
@misc{yang2026axioms,
title = {{Axioms of Continuous Separation}},
author = {Zhongqiang Yang and Nada Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.13086},
eprint = {2608.13086},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.13086},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Axioms of Continuous Separation
AU - Zhongqiang Yang
AU - Nada Wu
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2608.13086
UR - https://arxiv.org/abs/2608.13086
AB - For a T_1-space X, let Cld(X) denote all its nonempty closed subsets and T_4(X)={(F_1,F_2)∈ Cld(X)^2:F_1∩ F_2=∅}. In terms of closed sets, X is T_4 iff for each (F_1,F_2)∈ 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∩ ϕ_j(F_1,F_2)=∅ 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)→ 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?
ER -