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.

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.