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. The circle is a compact connected 1-manifold without boundary, while the real line is a noncompact connected example. A closed interval [0,1] is a compact 1-manifold with boundary, and an open interval is homeomorphic to R.

Classification results for 1-manifolds are simple: every connected 1-manifold without boundary is homeomorphic to R or to S^1. A connected compact 1-manifold without boundary is diffeomorphic to S^1. For manifolds with boundary, connected examples are homeomorphic to a closed interval [0,1], and more generally, manifolds with boundary decompose into finite disjoint unions of intervals.

Smooth structure: in dimension one, every 1-manifold admits a unique smooth structure, so the notions of topological and smooth 1-manifolds coincide up to diffeomorphism. Consequently, all connected 1-manifolds are orientable.

Applications and context: 1-manifolds model curves in geometry and topology. Embedded 1-manifolds in higher-dimensional spaces are studied as smooth curves, with relevance to fields such as knot theory and the theory of curves in manifolds.