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)

Comments

A fact related to the last paragraph above is Wallace's theorem (cf. [a1]): Every non-degenerate compact connected space contains at least two points that do not separate it.

References

[a1]	R. Engelking, "General topology" , Heldermann (1989)
[a2]	V. Guillemin, A. Pollace, "Differential topology" , Prentice-Hall (1974)
[a3]	D. Gale, "The classification of 1-manifolds: a take-home exam" Amer. Math. Monthly , 94 (1987) pp. 170–175