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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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