The Extended Real Line with Reentry: Separating US from KC in the Clontz Hierarchy

TL;DR

The paper introduces the Extended Real Line with Reentry (ERI), a new compact, path-connected, sober topological space that distinguishes US from KC and other separation axioms within the Clontz hierarchy.

Contribution

It constructs ERI, the first compact path-connected example separating US from KC, and analyzes its properties within the refined hierarchy of topological spaces.

Findings

ERI is compact, path-connected, sober, but not Hausdorff or KC.

ERI is at the $k_2$-Hausdorff level in Clontz's hierarchy.

The hierarchy level $k_2 ext{H}$-not-$ ext{wH}$ is invariant under certain topological operations.

Abstract

We construct the Extended Real Line with Reentry (ERI): identify $\{-\infty, 0, +\infty\}$ to a single point $\ast$ in $\overline{\mathbb{R}}$, and require every neighborhood of $\ast$ to have dense preimage. The resulting space is compact, path-connected, and sober; it is $T_1$ and US (uniquely sequential), but not weakly Hausdorff, not KC, and not Hausdorff. In the refined hierarchy of Clontz, ERI sits at the $k_2$-Hausdorff level. A search of pi-Base for compact US-not-KC spaces returns three entries -- $\mathbb{Q}^{\ast} \times \mathbb{Q}^{\ast}$, $\omega_1+1$ with doubled endpoint (S37), and the one-point compactification of the Arens-Fort space (S165) -- all totally disconnected. ERI is the first compact path-connected example. The same density condition on a general compact Hausdorff base without isolated points defines a Filter-Modified Quotient (FMQ). We prove that the density modifier $\mathcal{D}_Y$ is the least restrictive admissible modifier preserving US, and that the hierarchy level $k_2\mathrm{H}$-not-$\mathrm{wH}$ is invariant under infinite closed nowhere-dense collapse sets, iteration of the construction, and arbitrary products. The only remaining direction toward a US-not-$k_2\mathrm{H}$ level runs through non-first-countable base spaces.