Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Neighborhood"

m (Are "neighborhood" and "open ball" really used interchangeably in metric spaces? I'm skeptical.)
m
 
(One intermediate revision by one other user not shown)
Line 11: Line 11:
  
 
{{stub}}
 
{{stub}}
−
{{wikify}}
 
  
 +
[[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 $x$ 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.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Neighborhood&oldid=73420"