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Difference between revisions of "One-dimensional manifold"

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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.
 
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====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.W. Milnor,  "Topology from the differential viewpoint" , Univ. Virginia Press  (1965)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D.B. Fuks,  V.A. Rokhlin,  "Beginner's course in topology. Geometric chapters" , Springer  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.W. Hirsch,  "Differential topology" , Springer  (1976)</TD></TR></table>
 
 
 
  
 
====Comments====
 
====Comments====
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,  "General topology" , Heldermann  (1989)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V. Guillemin,  A. Pollace,  "Differential topology" , Prentice-Hall  (1974)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Gale,  "The classification of 1-manifolds: a take-home exam"  ''Amer. Math. Monthly'' , '''94'''  (1987)  pp. 170–175</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top"> J.W. Milnor,  "Topology from the differential viewpoint" , Univ. Virginia Press  (1965)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top"> D.B. Fuks,  V.A. Rokhlin,  "Beginner's course in topology. Geometric chapters" , Springer  (1984)  (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top">  M.W. Hirsch,  "Differential topology" , Springer  (1976)</TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,  "General topology" , Heldermann  (1989)</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  V. Guillemin,  A. Pollace,  "Differential topology" , Prentice-Hall  (1974)</TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Gale,  "The classification of 1-manifolds: a take-home exam"  ''Amer. Math. Monthly'' , '''94'''  (1987)  pp. 170–175</TD></TR>
 +
</table>
  
 
[[Category:Topology]]
 
[[Category:Topology]]

Latest revision as of 19:38, 25 March 2023

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.

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

[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)
[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=53236
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article