Epsilon Neighbourhood (Vector)

vector-calculus analysis

Definition

Epsilon Neighbourhood (Vector)

The $\epsilon$-neighbourhood of a vector ${\mathbf{x}}_0\in {\mathbb{R}}^n$ is the set of all vectors whose distance from ${\mathbf{x}}_0$ is strictly less than $\epsilon$:

$$U_{\epsilon }({\mathbf{x}}_0)=\{\mathbf{x}\in {\mathbb{R}}^n\mid \Vert \mathbf{x}-{\mathbf{x}}_0{\Vert }_2<\epsilon \}$$

where $\Vert \cdot {\Vert }_2$ denotes the Euclidean norm.

Geometrically, this set corresponds to an open ball centred at ${\mathbf{x}}_0$ with radius $\epsilon$. For $n=1$, it is an open interval; for $n=2$, it is an open disk; and for $n=3$, it is an open solid ball.

Significance

The epsilon neighbourhood is a fundamental tool for defining limits and continuity in multivariable calculus. It generalises the concept of proximity from the real line to higher-dimensional spaces. A set $S\subseteq {\mathbb{R}}^n$ is open if every point $\mathbf{x}\in S$ has an epsilon neighbourhood $U_{\epsilon }(\mathbf{x})$ that is contained entirely within $S$.