Definition:Deleted Neighborhood/Real Analysis

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Definition

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

Let $\map {N_\epsilon} \alpha$ be the $\epsilon$-neighborhood of $\alpha$:

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


Then the deleted $\epsilon$-neighborhood of $\alpha$ is defined as:

$\map {N_\epsilon} \alpha \setminus \set \alpha$.

That is, it is the $\epsilon$-neighborhood of $\alpha$ with $\alpha$ itself removed.


It can also be defined as:

$\map {N_\epsilon} \alpha \setminus \set \alpha : = \set {x \in \R: 0 < \size {\alpha - x} < \epsilon}$

or

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

from the definition of $\epsilon$-neighborhood.


Also known as

A deleted neighborhood is also called a punctured neighborhood.