The Extended Real Line with Reentry: A Compact Path-Connected US-not-KC Space

We construct what is, to the author's knowledge, the first compact path-connected US-not-KC space. The Extended Real Line with Reentry (ERI) identifies $-\infty$, $0$ and $+\infty$ to a single point $\ast$ of the extended real line and requires every neighborhood of $\ast$ to have dense preimage under the quotient map. ERI es compact, path-connected, sober, and Baire; it is $T_1$, US, and has unique limits even for transfinite sequences, but not KC, not weakly Hausdorff, and not Hausdorff; sequential convergence is explicit: $\hat{x}_n \to \ast$ iff $(x_n)$ has no accumulation point in $\mathbb{R}\setminus\{0\}$. The compact US-not-KC spaces previously on record are totally disconnected or fail path-connectedness; a one-point extension of $(0,1)$, analyzed alongside, achieves the same separation and is not homeomorphic to ERI.