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The subset of the closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017320/b0173201.png" /> of an (open) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017320/b0173202.png" />-dimensional real manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017320/b0173203.png" /> for which a neighbourhood of each point is homeomorphic to some domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017320/b0173204.png" /> in the closed half-space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017320/b0173205.png" />, the domain being open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017320/b0173206.png" /> (but not in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017320/b0173207.png" />). A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017320/b0173208.png" /> corresponding to a boundary point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017320/b0173209.png" />, i.e. to an intersection point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017320/b01732010.png" /> with the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017320/b01732011.png" />, is called a boundary point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017320/b01732012.png" />. A manifold having boundary points is known as a manifold with boundary. A compact manifold without boundary is known as a closed manifold. The set of all boundary points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017320/b01732013.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017320/b01732014.png" />-dimensional manifold without boundary.
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The subset of the closure  $  \overline{ {M  ^ {n} }}\; $
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of an (open)  $  n $-
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dimensional real manifold  $  M  ^ {n} $
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for which a neighbourhood of each point is homeomorphic to some domain  $  W  ^ {n} $
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in the closed half-space of  $  \mathbf R  ^ {n} $,
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the domain being open in  $  \mathbf R _ {+}  ^ {n} $(
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but not in  $  \mathbf R  ^ {n} $).
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A point  $  a \in \overline{ {M  ^ {n} }}\; $
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corresponding to a boundary point of  $  W  ^ {n} \subset  \mathbf R _ {+}  ^ {n} $,
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i.e. to an intersection point of  $  \overline{ {W  ^ {n} }}\; $
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with the boundary of  $  \mathbf R _ {+}  ^ {n} $,
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is called a boundary point of  $  M  ^ {n} $.
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A manifold having boundary points is known as a manifold with boundary. A compact manifold without boundary is known as a closed manifold. The set of all boundary points of  $  M  ^ {n} $
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is an  $  (n - 1) $-
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dimensional manifold without boundary.
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.W. Hirsch,  "Differential topology" , Springer  (1976)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.W. Hirsch,  "Differential topology" , Springer  (1976)</TD></TR></table>

Latest revision as of 06:28, 30 May 2020


The subset of the closure $ \overline{ {M ^ {n} }}\; $ of an (open) $ n $- dimensional real manifold $ M ^ {n} $ for which a neighbourhood of each point is homeomorphic to some domain $ W ^ {n} $ in the closed half-space of $ \mathbf R ^ {n} $, the domain being open in $ \mathbf R _ {+} ^ {n} $( but not in $ \mathbf R ^ {n} $). A point $ a \in \overline{ {M ^ {n} }}\; $ corresponding to a boundary point of $ W ^ {n} \subset \mathbf R _ {+} ^ {n} $, i.e. to an intersection point of $ \overline{ {W ^ {n} }}\; $ with the boundary of $ \mathbf R _ {+} ^ {n} $, is called a boundary point of $ M ^ {n} $. A manifold having boundary points is known as a manifold with boundary. A compact manifold without boundary is known as a closed manifold. The set of all boundary points of $ M ^ {n} $ is an $ (n - 1) $- dimensional manifold without boundary.

Comments

References

[a1] M.W. Hirsch, "Differential topology" , Springer (1976)
How to Cite This Entry:
Boundary (of a manifold). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boundary_(of_a_manifold)&oldid=46127
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article