Difference between revisions of "Boundary (of a manifold)"
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| + | 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==== | ====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=14973
Boundary (of a manifold). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boundary_(of_a_manifold)&oldid=14973
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article