Difference between revisions of "Neighborhood"
m (Are "neighborhood" and "open ball" really used interchangeably in metric spaces? I'm skeptical.)
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Revision as of 16:57, 28 March 2009
The neighborhood of a point is a notion which has slightly different meanings in different contexts. Informally, a neighborhood of in some space is a set that contains all points "sufficiently close" to . This notion may be formalized differently depending on the nature of the space.
Let be a metric space and let be an element of . A neighborhood of is the set of points in such that , for some positive real specific to . The real is called the radius of . This neighborhood is sometimes denoted . In metric spaces, neighborhoods are also called open balls.
Let be a topology, and let be an element of . We say that a set is a neighborhood of if there exists some open set for which .
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