A quasi-metrization theorem for hybrid topologies on the real line

Tom Richmond; Eliza Wajch

Home/Archive/arXiv

arXiv · 2307.16056

A quasi-metrization theorem for hybrid topologies on the real line

Tom Richmond·Eliza Wajch

View original source ↗Read source document ↗Export metadata ↓

Abstract

Hybrid topologies on the real line have been studied by various authors. Among the hybrid spaces, there are also Hattori spaces. However, some of the hybrid spaces are not homeomorphic to Hattori spaces. In this article, a common generalization of at least four kinds of the hybrid topologies on the real line is described. In the absence of the axiom of choice, a quasi-metrization theorem for such hybrid spaces is proved. It is shown that Kofner's quasi-metrization theorem for generalized ordered spaces is false in every model of $\mathbf{ZF}$ in which there exists an infinite Dedekind-finite subset of the real line.

Explore related subjects

math.GN

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Tom Richmond, Eliza Wajch. 2023-08-01. A quasi-metrization theorem for hybrid topologies on the real line. https://arxiv.org/abs/2307.16056

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

KEEP EXPLORING

Related papers

arXiv2026

Sobriety of Scott topologies under countability conditions

In this paper, we focus on the sobriety of the Scott topology in countable case. Specifically, we show that: (1) every meet-continuous core-compact countable dcpo is sober with respect to the Scott topology; (2)every countable locally compact dcpo is sober endowed with the Scott topology; (3) the lattice of all open sets for the rational numbers space Q equipped with the Scott topology is not sober.

math.GN↗

arXiv2026

Exponentiable Objects and Function spaces in Lowen Fuzzy Topological Spaces

We study exponentiable objects and function spaces in the category of stratified Lowen fuzzy topological spaces over $\I=[0,1]$. Using the Lowen fuzzy Sierpiński object $\Sier$, which identifies $τ_X$ with $C(X,\Sier)$, we explicitly determine the largest splitting topology on this mapping set. Its open weights $Φ:τ_X\to\I$ are precisely those satisfying Scott continuity and a finite-tier compatibility condition induced by finite powers of $\Sier$. This yields an intrinsic characterization: $X$ is exponentiable if and only if every $μ\inτ_X$ satisfies \[ μ=\bigvee_{λ\triangleleftΦ} (\const{Φ(μ)}\wedgeλ), \qquad λ\triangleleftΦ \Longleftrightarrow \const{Φ(ν)}\wedgeλ\leqν \quad(ν\inτ_X). \] When this condition holds, $Y^X$ has underlying set $C(X,Y)$, with topology generated by $[Φ,v](f)=Φ(v\circ f)$. We also obtain a dual closed-set formulation and three applications. Exponentiability implies that $τ_X$ is a continuous lattice, although the converse fails. Moreover, a classical space $X$ is exponentiable exactly when its induced fuzzy space $ωX$ is exponentiable in the entire stratified Lowen category. Finally, Lowen compact, strongly fuzzy compact, and $N$-compact Hausdorff spaces are exponentiable.

math.GN↗

arXiv2026

Super calibers in topological spaces and topological hyperspaces

We study the notion of a super caliber of a topological space, which is closely related to the classical notion of caliber and has appeared in the literature under several different names. We collect and unify several known results and establish new results concerning the collections of super calibers of topological spaces and their hyperspaces. In particular, we investigate the relationship between the super calibers of a space $X$ and those of hyperspaces $\mathcal{H}(X)$ lying between $\mathrm{CL}(X)$ and $\mathcal{F}(X)$. For infinite metrizable spaces, we characterize several cases in which $\mathsf{SC}(X)$ and $\mathsf{SC}(\mathrm{CL}(X))$ differ and establish an independence result over \textsf{ZFC}; see Theorem~5.12.

math.GN↗