boundary / frontier
Definition. Let be a topological space and let be a subset of . The boundary (or frontier) of is the set , where the overline denotes the closure of a set. Instead of , many authors use some other notation such as , , or . 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 , and .
The term ‘boundary’ (but not ‘frontier’) is used in a different sense for topological manifolds: the boundary of a topological -manifold is the set of points in that do not have a neighbourhood homeomorphic to . (Some authors define topological manifolds in such a way that they necessarily have empty boundary.) For example, the boundary of the topological -manifold is .
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 |