Epsilon Neighbourhood (Vector)

vector-calculus

Definition

Epsilon Neighbourhood (Vector)

The $\epsilon$-neighbourhood of a vector ${\mathbf{x}}_0\in {\mathbb{R}}^n$ is defined as:

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

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

This set is an interval for $n=1$, a circular surface for $n=2$, and a ball for $n=3$.