2026/03/03 by Damian Rafael Lattenero · 1 voice
Mathematics · #math.GN
arxiv published 2026/03/03 · arxiv updated 2026/04/22
We construct the Extended Real Line with Reentry (ERI): identify \-∞, 0, +∞\ to a single point ∗ in ℝ, and require every neighborhood of ∗ to have dense preimage. The resulting space is compact, path-connected, and sober; it is T1 and US (uniquely sequential), but not weakly Hausdorff, not KC, and not Hausdorff. In the refined hierarchy of Clontz, ERI sits at the k2-Hausdorff level. A search of pi-Base for compact US-not-KC spaces returns three entries -- ℚ∗ × ℚ∗, ω1+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 DY is the least restrictive admissible modifier preserving US, and that the hierarchy level k2H-not-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-k2H level runs through non-first-countable base spaces.