Neighborhood

A neighborhood of a point $x$ in a topological space $(X,\mathcal{T})$ is a subset $N\subseteq X$ such that there exists an open set $U\in\mathcal{T}$ with $x\in U\subseteq N$. Equivalently, $N$ is a neighborhood of $x$ if and only if $x\in \operatorname{int}(N)$, where interior is taken in $X$.

Neighborhoods give a point-based way to express concepts like limit points and continuity without referring directly to all open sets.

Examples:

• In $\mathbb{R}$ with the usual topology, $(x-\varepsilon,x+\varepsilon)$ is a neighborhood of $x$ for any $\varepsilon>0$.
• In a metric space , every open ball centered at $x$ is a neighborhood of $x$.
• In $\mathbb{R}$, the closed interval $[x-1,x+1]$ is a neighborhood of $x$ (it contains the open interval $(x-1,x+1)$).