Four problems about the irreducible determined $T_0$-topologies

Abstract

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.