A Minimal Non Hausdorff Counterexample in Covering Space Theory

Abstract

We construct a one dimensional, second countable, simply connected manifold that exhibits a single non Hausdorff fiber, sufficient to destroy the fundamental properties of classical covering space theory. The space, called the line with k inseparable origins, is defined by taking k copies of the real line and identifying all nonzero points across copies, so that each copy retains a distinct origin. These origins are T1 separated but not Hausdorff separated. We embed the punctured real line into a closed disk with a single accumulation point, and project the nonzero locus homeomorphically onto the embedded image. The projection map collapses all origins to the puncture point. Away from the singular point, the map is a local homeomorphism. At the singular point, however, the fiber is non Hausdorff: every neighborhood of one origin contains the others. As a consequence, path lifting and homotopy lifting fail, the monodromy representation is undefined, and the group of deck transformations is isomorphic to the symmetric group on k letters. Despite the total space being simply connected, the map cannot be classified as a covering map, branched cover, semicovering, or \'etale morphism. This provides a minimal dimensional, fully explicit example showing that the failure of Hausdorff separation at a single fiber suffices to break lifting properties and eliminate the usual Galois type correspondence between fundamental groups and deck transformations. It presents a sharp obstruction to any naive extension of covering space theory to non Hausdorff settings.