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

Abstract: We construct the Extended Real Line with Reentry (ERI), a quotient of the extended real line obtained by collapsing {-infinity, 0, +infinity} to a single point and imposing a density condition on neighborhoods of the collapsed point. 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 the collapsed point, identify the failure of first-countability as the mechanism enabling US without Hausdorff separation, and locate ERI in the refined hierarchy of 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 paper constructs the Extended Real Line with Reentry by collapsing −∞, 0, and +∞ and imposing a density-based neighborhood condition, producing a compact, path-connected, T₁ space that is US but not KC.
• The paper proves that a sequence converges to the collapsed point exactly when it has no accumulation point away from 0, establishing unique sequential limits despite non-unique net limits and failure of first-countability.
• The paper shows that compact subsets with interior are not closed, all continuous real-valued functions are constant, and the density-modified quotient generalizes to compact Hausdorff spaces without isolated points.

Overview and contribution

This paper constructs the Extended Real Line with Reentry (ERI), a quotient of the extended real line $\overline{\mathbb{R}} = [-\infty, +\infty]$ in which the three points $\{-\infty, 0, +\infty\}$ are collapsed to a single point $\ast$, equipped not with the standard quotient topology but with a strictly coarser topology imposing a density condition: a set $U$ is open only if, whenever $\ast \in U$, its preimage under the quotient map is dense in $\overline{\mathbb{R}}$. The resulting space is compact, $T_1$, path-connected, US (every convergent sequence has a unique limit), and not KC (some compact subset is not closed). It therefore provides an explicit witness to the strictness of the implication $\mathrm{KC} \Rightarrow \mathrm{US}$ in Wilansky's hierarchy $$T_2 \Rightarrow \mathrm{KC} \Rightarrow \mathrm{US} \Rightarrow T_1$$.

The paper's stated contribution is not merely the existence of a compact US-not-KC space—three such examples are already recorded in pi-Base—but the combination of properties those examples lack: ERI is path-connected, arises from an elementary explicit construction rather than MAD families or transfinite ordinals, admits a complete characterization of sequential convergence at $\ast$, and locates all non-Hausdorff phenomena in a single density condition.

The construction

ERI is defined on the quotient set $\{-\infty, 0, +\infty\}$0 where $\{-\infty, 0, +\infty\}$1 identifies exactly $\{-\infty, 0, +\infty\}$2. Its topology consists of sets $\{-\infty, 0, +\infty\}$3 whose preimage is open in $\{-\infty, 0, +\infty\}$4, subject to the additional requirement that preimages of neighborhoods of $\{-\infty, 0, +\infty\}$5 be dense. This is deliberately not the quotient topology: the standard quotient $\{-\infty, 0, +\infty\}$6 is compact Hausdorff, first-countable, and path-connected, so condition (b) is the sole mechanism destroying Hausdorffness, first-countability at $\{-\infty, 0, +\infty\}$7, and the KC property while preserving $\{-\infty, 0, +\infty\}$8 and US.

A structural lemma characterizes neighborhoods of $\{-\infty, 0, +\infty\}$9 completely: they are precisely the images $\ast$0 for closed nowhere-dense $\ast$1. From this follows the key intersection principle (KIP): every neighborhood of $\ast$2 meets every nonempty open set, i.e., every neighborhood of $\ast$3 is dense. The KIP immediately yields failure of Hausdorff ($\ast$4 cannot be separated from any other point), connectedness, and—via linear paths composed with $\ast$5—path-connectedness. Compactness holds because $\ast$6 refines no topology but is contained in the compact quotient topology $\ast$7; more precisely, coarsening a compact topology preserves compactness.

Sequential convergence and the US property

The paper's central technical result is a complete convergence criterion: for $\ast$8,

$\ast$9

The forward direction uses that a convergent subsequence together with its limit forms a countable closed, hence nowhere dense, set whose complement projects to a neighborhood of $U$0 eventually avoided by the sequence. The reverse direction uses Bolzano–Weierstrass: if $U$1 eventually left some neighborhood $U$2, infinitely many terms would lie in the compact set $U$3, producing an accumulation point in $U$4.

Notably, this criterion does not require $U$5 to converge in $U$6: distinct subsequences may tend to different elements of $U$7, yet all project to sequences converging to the single point $U$8. A case analysis then establishes uniqueness of sequential limits throughout the space, giving the US property. The paper contrasts this with the line with two origins, which is $U$9 and first-countable but not US ($\ast \in U$0 converges to both origins), and observes that $\ast \in U$1 alone does not suffice for unique sequential limits—the specific convergence criterion does the work.

At the net level, however, ERI behaves like a non-Hausdorff space: a standard filter-product construction produces a net converging to both $\ast \in U$2 and any chosen $\ast \in U$3. Consistently, ERI is not a sequential space: $\ast \in U$4 is sequentially closed but not closed. The gap between sequence-level uniqueness and net-level non-uniqueness exists precisely because first-countability fails at $\ast \in U$5.

Failure of KC and the role of first-countability

For any closed interval $\ast \in U$6, the image $\ast \in U$7 is compact (continuous image of a compact set) but not closed: $\ast \in U$8, yet $\ast \in U$9 fails the density condition since $\overline{\mathbb{R}}$0 has nonempty interior. Hence ERI is not KC. The failure is pervasive: $\overline{\mathbb{R}}$1 is compact-but-not-closed for every closed $\overline{\mathbb{R}}$2 with nonempty interior.

First-countability fails at $\overline{\mathbb{R}}$3 by a Baire category argument: a countable family of dense open preimages would have dense intersection, yielding a point $\overline{\mathbb{R}}$4 lying in every base element's preimage while excluded from the smaller neighborhood $\overline{\mathbb{R}}$5. The paper argues this failure is necessary, not incidental: by the classical equivalence (first-countable + US $\overline{\mathbb{R}}$6 $\overline{\mathbb{R}}$7), any first-countable US space is Hausdorff. Since ERI is first-countable at every point except $\overline{\mathbb{R}}$8, the single failure point is exactly what permits US without $\overline{\mathbb{R}}$9.

Position in refined hierarchies and functional triviality

Using Clontz's refined hierarchy and the Bella–Costantini chain $T_1$0, the paper shows ERI is SC (every convergent sequence with its limit forms a closed set) but not weakly Hausdorff (the continuous image $T_1$1 of compact Hausdorff $T_1$2 is not closed). Thus ERI witnesses strictness of $T_1$3 and occupies the same Clontz level as the one-point compactification of the Arens–Fort space, while being the only path-connected example there.

Two further results sharpen the picture. First, within the abstracted "filter-modified quotient" framework—which the paper develops generally, noting the density modifier corresponds to the double-negation Grothendieck topology—a Hausdorff barrier theorem proves that any FMQ space over a compact Hausdorff base with finite collapsed set, if KC, must be Hausdorff. Consequently the density modifier achieves the finest possible non-Hausdorff separation level; the barrier also follows classically from the fact that a quotient of compact Hausdorff is Hausdorff iff the quotient map is closed. Second, every continuous real-valued function on ERI is constant, so $T_1$4 and the Gelfand spectrum is a point: the topology supports nontrivial sequential convergence yet is too coarse for any continuous real-valued invariant to detect it.

Generalization

The construction extends verbatim: for any compact Hausdorff $T_1$5 without isolated points and finite nonempty $T_1$6, the density-modified collapse $T_1$7 is compact, $T_1$8, connected, US, and not KC. Connectedness of $T_1$9 is not required—the density condition forces connectedness even over disconnected bases—and path-connectedness does not generalize without further hypotheses. The generalized proof relies on the countable Baire lemma (countable subsets of compact Hausdorff spaces without isolated points have empty interior), which the paper identifies as the engine of the entire separation: sequences are countable (hence blocked from having second limits), while compact sets with interior have non-dense complements (hence fail to be closed).

Limitations and open questions

The paper concedes several boundaries of its results. Path-connectedness does not carry over to the general construction, so the generalization yields connected but generally not path-connected examples. The $\mathrm{KC} \Rightarrow \mathrm{US}$0-Hausdorff status of ERI is left undetermined; resolving it would either place ERI at the same Clontz level as the Arens–Fort one-point compactification or strictly below it. The net-with-two-limits example invokes the axiom of choice, and the comparison with van Douwen's space notes that the latter is KC whereas ERI is not, so ERI does not subsume that example's anti-Hausdorff strength. Finally, the triviality of $\mathrm{KC} \Rightarrow \mathrm{US}$1 means ERI, while topologically informative, carries no nonconstant continuous real-valued structure—an inherent constraint on any analytic application.

Conclusion

The paper delivers a compact, path-connected, explicitly constructed US-not-KC space with a complete convergence criterion, identifies failure of first-countability as the necessary mechanism enabling US without Hausdorff separation, situates the example precisely in the Clontz and Bella–Costantini refinements, and proves that within quotients of compact Hausdorff spaces the KC property forces Hausdorffness—so the density modifier attains the finest achievable non-Hausdorff level. The remaining open question is whether ERI is $\mathrm{KC} \Rightarrow \mathrm{US}$2-Hausdorff.