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Neighbourhood

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This is a draft article, under development. These unapproved articles are subject to a disclaimer.

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. \empty \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

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