boundary / frontier
Definition.
Let X be a topological space and let A be a subset
of X. The boundary (or frontier) of A is the set
∂A=ˉA∩¯X\A,
where the overline denotes the closure
of a set.
Instead of ∂A, many authors use some other notation
such as bd(A), fr(A), Ab or β(A).
Note that the ∂ symbol is also used for other meanings of ‘boundary’.
From the definition, it follows that the boundary of any set is a closed set. It also follows that ∂A=∂(X\A), and ∂X=∅=∂∅.
The term ‘boundary’ (but not ‘frontier’) is used in a different sense for topological manifolds: the boundary ∂M of a topological n-manifold M is the set of points in M that do not have a neighbourhood homeomorphic to ℝn. (Some authors define topological manifolds in such a way that they necessarily have empty boundary.)
For example, the boundary of the topological 1-manifold [0,1] is {0,1}.
Title | boundary / frontier |
---|---|
Canonical name | BoundaryFrontier |
Date of creation | 2013-03-22 13:34:46 |
Last modified on | 2013-03-22 13:34:46 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 17 |
Author | yark (2760) |
Entry type | Definition |
Classification | msc 54-00 |
Synonym | boundary |
Synonym | frontier |
Synonym | topological boundary |
Related topic | ExtendedBoundary |
Related topic | Interior |