Difference between revisions of "Neighborhood"
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| − | The '''neighborhood''' of a point is a notion which has slightly different meanings in different contexts. | + | The '''neighborhood''' of a point is a notion which has slightly different meanings in different contexts. Informally, a neighborhood of <math>x</math> in some space <math>X</math> is a [[set]] that contains all points "sufficiently close" to <math>x</math>. This notion may be formalized differently depending on the nature of the space. |
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| + | == Metric spaces == | ||
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| + | Let <math>X</math> be a [[metric space]] and let <math>x</math> be an [[element]] of <math>X</math>. A neighborhood <math>N</math> of <math>x</math> is the set of points <math>a</math> in <math>X</math> such that <math>d(a,x)<r</math>, for some positive real <math>r</math> specific to <math>N</math>. The real <math>r</math> is called the radius of <math>N</math>. This neighborhood is sometimes denoted <math>N_r(x)</math>. In metric spaces, neighborhoods are also called open balls. | ||
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== General topology == | == General topology == | ||
Let <math>X</math> be a [[topological space |topology]], and let <math>x</math> be an element of <math>X</math>. We say that a set <math>N \subset X</math> is a neighborhood of <math>x</math> if there exists some open set <math>S</math> for which <math>x \in S \subset N</math>. | Let <math>X</math> be a [[topological space |topology]], and let <math>x</math> be an element of <math>X</math>. We say that a set <math>N \subset X</math> is a neighborhood of <math>x</math> if there exists some open set <math>S</math> for which <math>x \in S \subset N</math>. | ||
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{{stub}} | {{stub}} | ||
| + | [[Category:Analysis]] | ||
[[Category:Topology]] | [[Category:Topology]] | ||
Latest revision as of 14:38, 1 December 2015
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.
Metric spaces
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.
General topology
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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