Editing Boundary (topology) (section)

== Boundary of a boundary ==

For any set <math>S, \partial S \supseteq \partial\partial S,</math> where [[List_of_mathematical_symbols_by_subject#Set_relations|<math>\,\supseteq\,</math>]] denotes the [[superset]] with equality holding if and only if the boundary of <math>S</math> has no interior points, which will be the case for example if <math>S</math> is either closed or open.  Since the boundary of a set is closed, <math>\partial \partial S = \partial \partial \partial S</math> for any set <math>S.</math>  The boundary operator thus satisfies a weakened kind of [[idempotence]].

In discussing boundaries of [[manifold]]s or [[simplex]]es and their [[simplicial complex]]es, 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.