Separation Axioms Among US
Steven Clontz, Marshall Williams
TL;DR
This work investigates separation axioms strictly between US and $k_2H$, introducing intermediate notions such as UR (unique radial convergence), UOK (unique one-point compactification extension), and UCR (unique C-radial convergence) and establishing the chain $k_2H \Rightarrow UOK \Rightarrow UCR \Rightarrow US$, with $T_2$ implying UR. It provides counterexamples to separate these properties (e.g., $\omega_1+1$ with a doubled endpoint is US but not $k_2H$) and demonstrates that UR does not imply $lH$, $sH$, or RC. A central contribution is a unified, general framework using C-generated spaces to reinterpret standard separation axioms as $\mathbf C$-Hausdorff properties, tying US/UR/UCR to $\mathbf S$-/$\mathbf R$-Hausdorff and $T_2$ to $\mathbf H$-Hausdorff, thereby offering a cohesive perspective that connects transfinite convergence, map-generated topologies, and diagonal-closedness across a spectrum of separation properties.
Abstract
A standard introductory result is that Hausdorff spaces have the property US, that is, each convergent sequence has a unique limit. This paper explores several existing and new characterizations of separation axioms that are strictly weaker than $T_2$ but strictly stronger than US.