# Boundary (of a manifold)

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