Definition:Open Neighborhood

From ProofWiki
Jump to navigation Jump to search

Definition

Let $T = \struct {S, \tau}$ be a topological space.

Let $A \subseteq S$ be a subset of $S$.

Let $N_A$ be a neighborhood of $A$.

Let:

$N_A \in \tau$

That is, let $N_A$ itself be an open set of $T$.


Then $N_A$ is called an open neighborhood of $A$ in $T$.


Open Neighborhood of Point

Let $x \in S$ be a point of $S$.

Let $N_x$ be a neighborhood of $x$ in $T$.

Let:

$N_x \in \tau$

That is, let $N_x$ itself be an open set of $T$.


Then $N_x$ is called an open neighborhood of $x$ in $T$.


Real Analysis

Real Numbers

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

Let $I \subseteq \R$ be a subset.


Then $I$ is an open neighborhood of $x$ if and only if $I$ is open and $I$ is a neighborhood of $x$.


Real Euclidean Space

Let $n\geq1$ be a natural number.

Let $x\in\R^n$.

Let $I \subseteq \R$ be a subset.


Then $I$ is an open neighborhood of $x$ if and only if $I$ is open and $I$ is a neighborhood of $x$.


Neighborhood defined as Open

Some authorities define a neighborhood of a set $A$ as what $\mathsf{Pr} \infty \mathsf{fWiki}$ defines as an open neighborhood:

$N_A$ is a neighborhood of $A$ if and only if $N_A$ is an open set of $T$ which itself contains $A$.

That is, in order to be a neighborhood of $A$ in $T$, $N_A$ must not only be a subset of $T$, but also be an open set of $T$.

However, this treatment is less common, and considered by many to be old-fashioned.

When the term neighborhood is used on this site, it is assumed to be not necessarily open unless so specified.


Also see


Linguistic Note

The UK English spelling of neighborhood is neighbourhood.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Open_Neighborhood&oldid=741937"