# One-dimensional manifold

A topological space \$X\$ each point of which has a neighbourhood homeomorphic to a line (an interior point) or to a half-line (a boundary point). A connected paracompact Hausdorff one-dimensional manifold without boundary points is homeomorphic to the circle if it is compact, and to the line if it is not compact; if one or two boundary points are present, then \$X\$ is homeomorphic to a half-open or closed bounded interval, respectively. Any such one-dimensional manifold can be smoothened, so that homeomorphic can be replaced by diffeomorphic in the assertions above.

A metric continuum (a connected compact metric space) \$K\$ of which each point, with two exceptions, separates it, is homeomorphic to a closed interval. If every two points separate \$K\$, then \$K\$ is homeomorphic to the circle. A subset \$A\subset K\$ separates \$K\$ if \$K\setminus A\$ can be written as a union of two open disjoint subsets.

#### References

 [1] J.W. Milnor, "Topology from the differential viewpoint" , Univ. Virginia Press (1965) [2] D.B. Fuks, V.A. Rokhlin, "Beginner's course in topology. Geometric chapters" , Springer (1984) (Translated from Russian) [3] M.W. Hirsch, "Differential topology" , Springer (1976)