Neighborhood - Knowlpedia

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

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 $x\in U\subseteq N$ .

Equivalent characterizations

Equivalently, $N$ is a neighborhood of $x$ if and only if $x\in \operatorname{int}(N)$ , where interior is taken in $X$ .

Remarks

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 $(x-1,x+1)$ ).