Definition:Neighborhood (Metric Space)/Compact Subset

This page is about neighborhood in the context of metric spaces. For other uses, see neighborhood.

Definition

Let $M = \struct {A, d}$ be a metric space.

Let $K \subseteq A$ be a compact subset of $A$.

The $\epsilon$-neighborhood of $K$ in $M$ defined and denoted as:

$\map {\NN_\epsilon} K := \set {x \in A: \exists y \in K: \map d {x, y} \le \epsilon}$

Also see

• Results about neighborhoods can be found here.

Linguistic Note

The UK English spelling of neighborhood is neighbourhood.

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Hausdorff metric