Definition:Neighborhood (Real Analysis)

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This page is about neighborhoods in the real number line. For other uses, see Definition: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:

$N_\epsilon \left({\alpha}\right) := \left({\alpha - \epsilon \,.\,.\, \alpha + \epsilon}\right)$

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


Linguistic Note

The UK English spelling of neighborhood is neighbourhood.