Publication Date

Spring 2026

Degree Type

Thesis

Degree Name

Master of Arts (MA)

Department

Mathematics and Statistics

Advisor

Edgar Bering IV; Jordan Schettler; Slobodan Simic

Abstract

In this paper we seek to explain the relationship between foliations of the plane, Kaplan diagrams, and simply connected non-Hausdorff 1-manifolds with countable basis that are orientable with an ordering on branch points. We will walk the reader through the definitions of all of these objects, and provide examples with a focus on the motivating example of the Reeb foliation. This paper will describe and define the known bijection from the set of foliations of the plane, F, to the set of Kaplan diagrams, K, its inverse, and the one-to-one map from F to the set of simply connected non-Hausdorff 1-manifolds with countable basis that are orientable with an ordering on branch points, V. Finally, we will introduce and define new bijections: ˜φ : K → V, ˜Δ : V → K, and p : V → F. These bijections will establish how to map Kaplan diagrams to corresponding non-Hausdorff manifolds, how to map back to a Kaplan diagram from the 1-manifold, and provide the missing proof that the one-to-one map from F to the set of orientable simply connected 1-dimensional non-Hausdorff manifolds with an ordering on the branch points is indeed a bijection.

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