# Definition:Neighborhood (Topology)/Set

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## Definition

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

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

A **neighborhood** of $A$, which can be denoted $N_A$, is any subset of $S$ containing an open set of $T$ which itself contains $A$.

That is:

- $\exists U \in \tau: A \subseteq U \subseteq N_A \subseteq S$

## Also defined as

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.

## Linguistic Note

The UK English spelling of **neighborhood** is **neighbourhood**.

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction