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Neighborhood: Difference between revisions

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definition
 
m Are "neighborhood" and "open ball" really used interchangeably in metric spaces? I'm skeptical.
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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.
 
== Metric spaces ==
 
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.
 


== 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>.
== Metric spaces ==
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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[[Category:Topology]]
[[Category:Topology]]

Revision as of 10:02, 29 June 2008

The neighborhood of a point is a notion which has slightly different meanings in different contexts. Informally, a neighborhood of $x$ in some space $X$ is a set that contains all points "sufficiently close" to $x$. This notion may be formalized differently depending on the nature of the space.

Metric spaces

Let $X$ be a metric space and let $x$ be an element of $X$. A neighborhood $N$ of $x$ is the set of points $a$ in $X$ such that $d(a,x)<r$, for some positive real $r$ specific to $N$. The real $r$ is called the radius of $N$. This neighborhood is sometimes denoted $N_r(x)$. In metric spaces, neighborhoods are also called open balls.


General topology

Let $X$ be a topology, and let $x$ be an element of $X$. We say that a set $N \subset X$ is a neighborhood of $x$ if there exists some open set $S$ for which $x \in S \subset N$.

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