# 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.

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