Neighbourhood (topology)

In topology, a neighbourhood of a point x in a topological space X is a set N such that x is in the interior of N; that is, there is an open set U such that $x\in U\subseteq N$. A neighbourhood of a set A in X is a set N such that A is contained in the interior of N; that is, there is an open set U such that $A\subseteq U\subseteq N$.

The family of neighourhoods of a point x, denoted ${\mathcal {N}}_{x}$ satisfies the properties

1. $X\in {\mathcal {N}}_{x};\,$
2. $\emptyset \not \in {\mathcal {N}}_{x};\,$
3. $U,V\in {\mathcal {N}}_{x}\Rightarrow U\cap V\in {\mathcal {N}}_{x};\,$
4. $U\in {\mathcal {N}}_{x}{\mbox{ and }}U\subseteq N\Rightarrow N\in {\mathcal {N}}_{x}.\,$

The properties are equivalent to stating that the neighbourhood system ${\mathcal {N}}_{x}$ is a filter, the neighbourhood filter of x.

A topology may be defined in terms of its neighbourhood systems: a set is open if and only if it is a neighbourhood of each of its points.

See also

• Topological space#Some topological notions