Neighborhood

From AoPSWiki

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 $x \in S \subset N$ .

This article is a stub. Help us out by expanding it.
This article has been found by the AoPSWiki Editors not to be written in wiki style. Help us out by putting this page in wiki style and removing this message.

Personal tools

Navigation

Search

Toolbox

Neighborhood

From AoPSWiki

Metric spaces

General topology