The dynamical topics have been left outside the scope of this text. That's why most the results below are provided with proofs.īesides the adjacent field of dynamics, there is no research activity related to the topology of 1-manifolds. The proofs are really elementary, but use ideas that are barely applicable in similar high-dimensional situations. It was difficult to find appropriate references. This happened, probably, because it is considered too simple. The material presented below is not well represented in the literature. To be on the safe side, we use an unambiguous term manifold of dimension or -manifold. Even in the context of topology, the term curve may mean not only a manifold of dimension 1 with an additional structure, but, for instance, an immersion of a smooth manifold of dimension 1 to Euclidean space. Manifolds of dimension 1 are called curves, but this name may lead to a confusion, because many mathematical objects share it. 10.8 Sheaf of germs of functions on a 1-manifoldĪccording to the general definition of manifold, a manifold of dimension 1 is a topological space which is second countable (i.e., its topological structure has a countable base), satisfies the Hausdorff axiom (any two different points have disjoint neighborhoods) and each point of which has a neighbourhood homeomorphic either to the real line or to the half-line.10.7 Relatives of 1-manifolds without countable base.10.6 Homeomorphic, but non-diffeomorphic non-Hausdorff 1-manifolds.10.4 Non-orientable non-Hausdorff 1-manifolds.10.3 Uncountable family of non-homeomorphic connected non-Hausdorff 1-manifolds.10.1 First examples of non-Hausdorff 1-manifolds.8.2 Homotopy types of groups of homeomorphisms.3.7 Characterizing connected 1-manifolds in terms of separating points.3.4 About proofs of the classification theorems.3.3 Characterizing the topological type of a connected 1-manifold.
3.2 Topological classification of connected 1-manifolds.3.1 Reduction to classification of connected manifolds.
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