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One-dimensional manifold

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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
How to Cite This Entry:
One-dimensional manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=One-dimensional_manifold&oldid=33624
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article