Difference between revisions of "Boundary"
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| − | + | In [[topology]], the '''boundary''' of a [[set]] ''X'' is the [[intersection]] of the [[closure]] of ''X'' and the closure of the [[complement]] of ''X''. | |
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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]]. | ||
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| + | [[category:topology]] | ||
Revision as of 01:24, March 1, 2012
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.