Definition:Neighborhood (Topology)

Definition

Let $\left({X, \vartheta}\right)$ be a topological space.

Neighborhood of a Set

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

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

That is:

$\exists U \in \vartheta: A \subseteq U \subseteq N_A \subseteq X$

Neighborhood of a Point

The set $A$ can be a singleton, in which case the definition is of the neighborhood of a point.

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

A neighborhood of $z$, which can be denoted $N_z$, is any subset of $X$ containing an open set which itself contains $z$.

That is:

$\exists U \in \vartheta: z \in U \subseteq N_z \subseteq X$

Open Neighborhood

If $N_A \in \vartheta$, i.e. if $N_A$ is itself open in $X$, then $N_A$ is called an open neighborhood.

Some authorities require all neighborhoods to be open.

Closed Neighborhood

If $\complement_X \left({N_A}\right) \in \vartheta$, i.e. if $N_A$ is closed in $X$, then $N_A$ is called a closed neighborhood.

Elementary Properties

• From this definition, it follows directly that $X$ itself is always a neighborhood of any $A \subseteq X$.

• It also follows that any open set of $X$ containing $A$ is a neighborhood of $A$.

A set which is the neighborhood of all its points is open.

Linguistic Note

The UK English spelling of this is neighbourhood.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$