boundary / frontier

Definition. Let X be a topological spaceMathworldPlanetmath and let A be a subset of X. The boundary (or frontier) of A is the set A=A¯X\A¯, where the overline denotes the closurePlanetmathPlanetmath 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 manifoldsMathworldPlanetmathPlanetmath: the boundary M of a topological n-manifold M is the set of points in M that do not have a neighbourhood homeomorphic to n. (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