Boundary (topology)

A connected component of the boundary of S is called a boundary component of S.

There are several equivalent definitions for the boundary of a subset S of a topological space X:

These last two examples illustrate the fact that the boundary of a dense set with empty interior is its closure.

In discussing boundaries of manifolds or simplexes and their simplicial complexes, one often meets the assertion that the boundary of the boundary is always empty. Indeed, the construction of the singular homology rests critically on this fact. The explanation for the apparent incongruity is that the topological boundary (the subject of this article) is a slightly different concept from the boundary of a manifold or of a simplicial complex. For example, the boundary of an open disk viewed as a manifold is empty, as is its topological boundary viewed as a subset of itself, while its topological boundary viewed as a subset of the real plane is the circle surrounding the disk. Conversely, the boundary of a closed disk viewed as a manifold is the bounding circle, as is its topological boundary viewed as a subset of the real plane, while its topological boundary viewed as a subset of itself is empty. (In particular, the topological boundary depends on the ambient space, while the boundary of a manifold is invariant.)