A continuous $3$-distributive frame that is not $ω$-distributive

Wei Luan; Qingguo Li

Home/Archive/arXiv

arXiv · 2609.11671

A continuous $3$-distributive frame that is not $ω$-distributive

Wei Luan·Qingguo Li

View original source ↗Read source document ↗Export metadata ↓

Abstract

We give a negative answer to the question, posed by Erné, whether every $3$-distributive lattice is $ω$-distributive. More precisely, we exhibit a continuous frame that is $κ$-distributive for every integer $κ\geq 2$, but is not a wide coframe. The frame is the open-set lattice of a compact, locally compact, countably based $T_0$ topological meet-semilattice, obtained from Lawson's construction in the logarithmic form described by Goubault-Larrecq. The failure of $ω$-distributivity is witnessed by an explicit matrix with countably many nonempty finite rows: all row joins are the same nonzero element, whereas every choice of one entry from each row has meet zero. The same space answers negatively Erné's accompanying question whether every $4$-web space is a wide web space. All properties of the construction needed for these conclusions are proved directly.

Explore related subjects

math.GN

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Wei Luan, Qingguo Li. 2026-09-10. A continuous $3$-distributive frame that is not $ω$-distributive. https://arxiv.org/abs/2609.11671

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The circle as a topological fractal

We prove that no family of two continuous self-maps witnesses that the circle is a topological fractal, answering a question of Karasová and the present author. Since three maps are known to suffice, this bound is optimal. In contrast, for every $\varepsilon>0$ there are two continuous self-maps of the circle, depending on $\varepsilon$, whose images cover the circle and an integer $N$ such that every composition of $N$ of them has image of diameter less than $\varepsilon$. Thus two maps suffice at any prescribed scale, but no fixed pair works at all scales.

math.GN↗

arXiv2026

Finite-Point Metrizable Coarsenings: Compatible Gauges, Simplicial Metrics, and Hausdorff Lower Bounds

Let $(X,τ)$ be metrizable and let $F=\{a_1,\ldots,a_k\}\subseteq X$, where $2\le k<\infty$. We represent all metrizable topologies $σ\subseteqτ$ agreeing with $τ$ on $X\setminus F$ by compatible systems of continuous gauges $s_i:X\to[0,1]$ with $s_i^{-1}(0)=\{a_i\}$. The condition $\inf_X\max\{s_i,s_j\}>0$ for $i\ne j$ is equivalent to both Hausdorffness and metrizability of the prescribed gauge topology. A normalized product map into the standard simplex gives an explicit metric; its triangle inequality follows from a simplex slack inequality. This metric is complete whenever the auxiliary bounded compatible metric is complete. For two compatible systems, their coordinatewise minimum describes the intersection topology. It is compatible exactly when the two coarsenings have a common Hausdorff lower bound; in that case the intersection is metrizable and is their meet. Otherwise every common lower topology is non-Hausdorff. A closed-discrete construction produces such an obstructed pair for every noncompact metrizable space and every finite exceptional set with at least two points. Consequently, for these exceptional sets, the family is downward directed, or is a lattice, if and only if $(X,τ)$ is compact, in which case it consists only of $τ$.

math.GN↗

arXiv2026

Journey into special $T_1$-spaces

In this survey, we review certain types of special $T_{1}$ spaces and their associated fixed-point theorems. Kupka introduced the notion of a feeble topological contraction, which naturally generalizes Lipschitz contractions defined on metric spaces. Specifically, Kupka established a fixed-point theorem for feeble topological contractions possessing a closed graph within the product of arbitrary $T_{0}$ spaces. Furthermore, the $T_{1}$ separation axiom is shown to guaranty the uniqueness of such fixed points. Finally, we discuss peripheral Hausdorff and locally Hausdorff spaces within the context of these fixed-point results.

math.GN↗