Four problems about the irreducible determined T0-topologies
Abstract
We answer four questions on irreducible determined T0-topologies and a related question on monotone determined topologies posed by Xu, Yang and Luo (Topology Appl.\ 382 (2026), 109762). For every T0 topology τ, the family τr used to define its irreducible-derived topology is already a topology and preserves all nonempty irreducible subsets and their closures. Consequently, the least irreducible determined refinement is obtained in one step: τr=rd(τ)=Rd(τ). For an arbitrary poset P, we describe the weakest order-compatible irreducible determined topology ρ(P) using Dedekind--MacNeille cuts and a finite-cover condition involving principal lower sets. The Scott space of the binary-prefix domain provides a common counterexample to saturated heredity for irreducible determined spaces and monotone determined spaces: its maximal-point subspace is Cantor space. We also describe ρ(P)-closure by iterating an irreducible-cut operator, characterize Scott continuity of this operator, and give an example requiring two applications of the cut operator. A finite-decomposition condition extends the topology and closure-preservation results to general subset systems. For the Rudin and well-filtered determined systems, we exhibit a common T1 example whose first and second derived topologies are the co-countable and discrete topologies, respectively. Thus the first derived topology need not be determined by the same subset system.
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BibTeX
@misc{zhao2026four,
title = {{Four problems about the irreducible determined $T_0$-topologies}},
author = {Chongyun Zhao and Xiaoquan Xu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04808},
eprint = {2611.04808},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04808},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Four problems about the irreducible determined T_0-topologies
AU - Chongyun Zhao
AU - Xiaoquan Xu
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04808
UR - https://arxiv.org/abs/2611.04808
AB - We answer four questions on irreducible determined T_0-topologies and a related question on monotone determined topologies posed by Xu, Yang and Luo (Topology Appl.\ 382 (2026), 109762). For every T_0 topology τ, the family τ_r used to define its irreducible-derived topology is already a topology and preserves all nonempty irreducible subsets and their closures. Consequently, the least irreducible determined refinement is obtained in one step: τ_r=rd(τ)=Rd(τ). For an arbitrary poset P, we describe the weakest order-compatible irreducible determined topology ρ(P) using Dedekind--MacNeille cuts and a finite-cover condition involving principal lower sets. The Scott space of the binary-prefix domain provides a common counterexample to saturated heredity for irreducible determined spaces and monotone determined spaces: its maximal-point subspace is Cantor space. We also describe ρ(P)-closure by iterating an irreducible-cut operator, characterize Scott continuity of this operator, and give an example requiring two applications of the cut operator. A finite-decomposition condition extends the topology and closure-preservation results to general subset systems. For the Rudin and well-filtered determined systems, we exhibit a common T_1 example whose first and second derived topologies are the co-countable and discrete topologies, respectively. Thus the first derived topology need not be determined by the same subset system.
ER -