# Definition:Neighborhood (Topology)/Point

## Definition

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

Let $z \in S$ be a point in a $S$.

Let $N_z$ be a subset of $S$ which contains (as a subset) an open set of $T$ which itself contains (as an element) $z$.

Then $N_z$ is a **neighborhood** of $z$.

That is:

- $\exists U \in \tau: z \in U \subseteq N_z \subseteq S$

## 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

- Results about
**neighborhoods**can be found here.

## Linguistic Note

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

## Sources

- 1962: Bert Mendelson:
*Introduction to Topology*... (previous) ... (next): $\S 3.2$: Topological Spaces: Definition $2.2$ - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.7$: Definitions: Definition $3.7.5$ - 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