1manifold
A 1-manifold, commonly written as a 1-manifold, is a topological manifold of dimension one. Concretely, it is a Hausdorff and second-countable space in which every point has a neighborhood homeomorphic to an open interval of the real line. If a space permits neighborhoods homeomorphic to half-open intervals, it is called a 1-dimensional manifold with boundary; otherwise it is a 1-manifold without boundary.
Examples include the real line R, the circle S^1, and finite or infinite disjoint unions of these.
Classification results for 1-manifolds are simple: every connected 1-manifold without boundary is homeomorphic to R or
Smooth structure: in dimension one, every 1-manifold admits a unique smooth structure, so the notions of topological
Applications and context: 1-manifolds model curves in geometry and topology. Embedded 1-manifolds in higher-dimensional spaces are