this same point: However, the topological boundary and topological closure in of the open unit ball are: In particular, the open unit ball's topological boundary is a subset of the unit sphere in And the open unit ball's topological closure is a proper subset of the closed unit ball in The point for instance, cannot belong to because there does not exist a sequence in that converges to it; the same reasoning generalizes to also explain why no point in outside of the closed sub-interval belongs to Because the topological boundary of the set is always a subset of 's closure, it follows that must also be a subset of In any metric space the topological boundary in of an open ball of radius centered at a point is always a subset of the sphere of radius centered at that same point ; that is, always holds. Moreover, the unit sphere in contains which is an open subset of This shows, in particular, that the unit sphere in contains a subset of Boundary of a boundary For any set where denotes the superset with equality holding if and only if the boundary of has no interior points, which will be the case for example if is either closed or open. Since the boundary of a set is closed, for any set The boundary operator thus satisfies a weakened kind of idempotence. 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