In this work, we will explore fundamental concepts of general topology, manifolds, and foliations, with the central goal of demonstrating the main result presented in the article by André Haefliger and George Reeb [3]. This result establishes that the leaf space of a foliation of the plane has the structure of a one-dimensional manifold, possibly non-Hausdorff. Initially, we will introduce basic concepts of general topology, with an emphasis on quotient topology, providing essential examples for understanding the subject. Subsequently,we delve into the study of topological manifolds, including concepts such as the quotient space of a manifold, simply connected spaces, and examples focused on one-dimensional manifolds. We will pay special attention to the definition of foliations on a manifold, now assumed to be Hausdorff, and their equivalence classes known as leaves. Several results about leaf spaces and examples of foliations will be presented. We will explore the intimate relationship between plane foliations and the structure of simply connected one-dimensional manifolds. Finally,we highlight how these results provide insights into a proof of the Kaplan Theorem.
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