# 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.

#### 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