Definition:Neighborhood (Real Analysis)

This page is about neighborhood in the context of real analysis. For other uses, see neighborhood.

Definition

Let $\alpha \in \R$ be a real number.

Open Subset Neighborhood

Let $N_\alpha$ be a subset of $\R$ which contains (as a subset) an open real set which itself contains (as an element) $\alpha$.

Then $N_\alpha$ is a neighborhood of $\alpha$.

Epsilon-Neighborhood

On the real number line with the usual metric, the $\epsilon$-neighborhood of $\alpha$ is defined as the open interval:

$\map {N_\epsilon} \alpha := \openint {\alpha - \epsilon} {\alpha + \epsilon}$

where $\epsilon \in \R_{>0}$ is a (strictly) positive real number.

Linguistic Note

The UK English spelling of neighborhood is neighbourhood.