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The boundary of an -manifold with boundary is an ()-manifold. A disk (circle plus interior) is a 2-manifold with boundary. Its boundary is a circle, a 1-manifold. A square with interior is also a 2-manifold with boundary. A ball (sphere plus interior) is a 3-manifold with boundary. Its boundary is a sphere, a 2-manifold.
A boundary point of a set is any element of that set's boundary. The boundary defined above is sometimes called the set's topological boundary to distinguish it from other similarly named notions such as the boundary of a manifold with boundary or the boundary of a manifold with corners, to name just a few examples.
But normally, a compact manifold (compact with respect to its underlying topology) can synonymously be used for closed manifold if the usual definition for manifold is used. The notion of a closed manifold is unrelated to that of a closed set. A line is a closed subset of the plane, and it is a manifold, but not a closed manifold.
The surface S is said to be boundary-compressible if either S is a disk that cobounds a ball with a disk in or there exists a boundary-compressing disk for S in M. Otherwise, S is boundary-incompressible. Alternatively, one can relax this definition by dropping the requirement that the surface be properly embedded.
The boundary of an (+)-dimensional manifold is an -dimensional manifold that is closed, i.e., with empty boundary. In general, a closed manifold need not be a boundary: cobordism theory is the study of the difference between all closed manifolds and those that are boundaries.
A manifold is called a "k-handlebody" if it is the union of r-handles, for r at most k. This is not the same as the dimension of the manifold. For instance, a 4-dimensional 2-handlebody is a union of 0-handles, 1-handles and 2-handles. Any manifold is an n-handlebody, that is, any manifold is the union of handles.
A manifold admits a nowhere-vanishing volume form if and only if it is orientable. An orientable manifold has infinitely many volume forms, since multiplying a volume form by a nowhere-vanishing real valued function yields another volume form. On non-orientable manifolds, one may instead define the weaker notion of a density.
The boundary of a manifold is a manifold , which has dimension . An orientation on M {\displaystyle M} induces an orientation on ∂ M {\displaystyle \partial M} . We usually denote a submanifold by Σ ⊂ M {\displaystyle \Sigma \subset M} .