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The Extended Real Line with Reentry: A Compact Quotient Space Separating US from KC

Damian Rafael Lattenero

Abstract

We construct the \emph{Extended Real Line with Reentry} (ERI), a quotient of $\overline{\mathbb{R}} = [-\infty,+\infty]$ obtained by collapsing $\{-\infty, 0, +\infty\}$ to a single point $\ast$ and imposing a density condition on neighborhoods of~$\ast$. ERI is compact, $T_1$, path-connected, US, and not~KC, providing an explicit, constructive example separating US from~KC in the Wilansky hierarchy. We give a complete convergence criterion at~$\ast$, identify the failure of first-countability as the mechanism enabling US without Hausdorff separation, and locate ERI in the refined hierarchy of Clontz~\cite{bib:clontz}: ERI is SC (sequentially closed) but not weakly Hausdorff. The construction generalizes to arbitrary compact Hausdorff spaces without isolated points, and the only continuous real-valued functions on ERI are the constants.

The Extended Real Line with Reentry: A Compact Quotient Space Separating US from KC

Abstract

We construct the \emph{Extended Real Line with Reentry} (ERI), a quotient of obtained by collapsing to a single point and imposing a density condition on neighborhoods of~. ERI is compact, , path-connected, US, and not~KC, providing an explicit, constructive example separating US from~KC in the Wilansky hierarchy. We give a complete convergence criterion at~, identify the failure of first-countability as the mechanism enabling US without Hausdorff separation, and locate ERI in the refined hierarchy of Clontz~\cite{bib:clontz}: ERI is SC (sequentially closed) but not weakly Hausdorff. The construction generalizes to arbitrary compact Hausdorff spaces without isolated points, and the only continuous real-valued functions on ERI are the constants.
Paper Structure (35 sections, 33 theorems, 5 equations)

This paper contains 35 sections, 33 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. The Wilansky hierarchy
  3. Known examples
  4. Contribution
  5. The ERI Construction
  6. Definition and notation
  7. Preliminary facts
  8. Density facts
  9. Topology axioms and basic properties
  10. Neighborhoods of *
  11. The key intersection principle
  12. Separation and Topological Properties
  13. Countability and Sequential Properties
  14. First countability fails at *
  15. Sequential convergence
  16. ...and 20 more sections

Key Result

Lemma 2.4

$\overline{\mathbb{R}}$ has no isolated points.

Theorems & Definitions (87)

  • Definition 1.1: Wilansky bib:wilansky
  • Definition 2.1: ERI space
  • Remark 2.2: Standing notation
  • Remark 2.3
  • Lemma 2.4: No isolated points
  • proof
  • Corollary 2.5
  • proof
  • Definition 2.6: Saturated set
  • Lemma 2.7
  • ...and 77 more