Neighborhood

Let $(X,d)$ be a metric space and let $x\in X$. A set $N\subseteq X$ is a neighborhood of $x$ if there exists $r>0$ such that

(see open ball ).

Neighborhoods encode the local structure around a point. Many definitions in analysis can be phrased using neighborhoods (e.g., limit points , closure , continuity ).

Examples:

• In $\mathbb{R}$, any interval of the form $(x-\varepsilon,x+\varepsilon)$ is a neighborhood of $x$.
• In $\mathbb{R}$, the set $[x-1,x+1]$ is a neighborhood of $x$ (it contains the open ball $(x-1,x+1)$).
• In a discrete metric space, $\{x\}$ is a neighborhood of $x$ (since $B(x,1)=\{x\}$).