Difference between revisions of "Boundary"
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Every set has a boundary, but only [[closed set]]s contain their boundaries. In contrast, an [[open set]] never contains its boundary. Example: both [0,1] and (0,1) have the boundary {0,1}. The first set is closed and contains its boundary; the second set is open and does not contain its boundary. Depending on the context, boundaries can be referred to by more common names: [[perimeter]], [[surface]], [[frontier]], [[border]], and [[edge]]. | Every set has a boundary, but only [[closed set]]s contain their boundaries. In contrast, an [[open set]] never contains its boundary. Example: both [0,1] and (0,1) have the boundary {0,1}. The first set is closed and contains its boundary; the second set is open and does not contain its boundary. Depending on the context, boundaries can be referred to by more common names: [[perimeter]], [[surface]], [[frontier]], [[border]], and [[edge]]. | ||
| − | [[ | + | [[Category:Topology]] |
Latest revision as of 03:34, July 13, 2016
In topology, the boundary of a set X is the intersection of the closure of X and the closure of the complement of X.
Every set has a boundary, but only closed sets contain their boundaries. In contrast, an open set never contains its boundary. Example: both [0,1] and (0,1) have the boundary {0,1}. The first set is closed and contains its boundary; the second set is open and does not contain its boundary. Depending on the context, boundaries can be referred to by more common names: perimeter, surface, frontier, border, and edge.