Neighborhoods: Delta-Neighborhoods & Deleted Neighborhoods

A neighborhood represents a symmetric open interval centered around a fixed point. Neighborhoods and deleted neighborhoods form the geometric and analytical backbone of limit definitions ( $ϵ$ - $δ$ proofs) in calculus.

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Term	Notation	Inequality	Interval Representation
$δ$ -Neighborhood	$N (a; δ)$	$\| x - a \| < δ$	$(a - δ, a + δ)$
Deleted $δ$ -Neighborhood	$N^{*} (a; δ)$	$0 < \| x - a \| < δ$	$(a - δ, a) \cup (a, a + δ)$

𝛅-Neighborhoods

Let I be the set of all points whose distance from a fixed point a is less than a number $δ > 0$ . Then

\begin{aligned} I &= \{x \mid |x-a| < \delta\} \\ &= \{x \mid -\delta < x-a < \delta\} &&{\small (|t|

Such a set is called a neighborhood (or, more precisely, the $δ$ -neighborhood) of a and $δ$ is called the radius of the neighborhood. The $δ$ -neighborhood of a is shown in the following figure.

TikZ figure — The $δ$ -neighborhood of a is the set of all points whose distance from a is less than $δ > 0$ . The radius of this neighborhood is $δ$ .

We denote neighborhoods by $N (a), N_{1} (a), N_{2} (a), \dots$ . To indicate that the radius of the neighborhood is $δ$ , we may write $N (a; δ)$ .

Deleted Neighborhoods

The open interval centered at a with radius $δ$ , with the point a itself excluded or "deleted" from the interval is called the deleted $δ$ -neighborhood of a. In other words, the deleted $δ$ -neighborhood of a is

{x ∣ | x - a | < δ, x \neq a}

{x ∣ 0 < | x - a | < δ}

The deleted $δ$ -neighborhood of a is shown in the following figure.

Neighborhoods

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𝛅-Neighborhoods

Deleted Neighborhoods

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