Glossary of Topology - N

N

Neighbourhood/Neighborhood

A neighbourhood of a point x is a set containing an open set which in turn contains the point x. More generally, a neighbourhood of a set S is a set containing an open set which in turn contains the set S. A neighbourhood of a point x is thus a neighbourhood of the singleton set {x}. (Note that under this definition, the neighbourhood itself need not be open. Many authors require that neighbourhoods be open; be careful to note conventions.)

Neighbourhood base/basis

See Local base.

Neighbourhood system for a point x

A neighbourhood system at a point x in a space is the collection of all neighbourhoods of x.

Net

A net in a space X is a map from a directed set A to X. A net from A to X is usually denoted (x_α), where α is an index variable ranging over A. Every sequence is a net, taking A to be the directed set of natural numbers with the usual ordering.

Normal

A space is normal if any two disjoint closed sets have disjoint neighbourhoods. Every normal space admits a partition of unity.

Normal Hausdorff

A normal Hausdorff space (or T₄ space) is a normal T₁ space. (A normal space is Hausdorff if and only if it is T₁, so the terminology is consistent.) Every normal Hausdorff space is Tychonoff.

Nowhere dense

A nowhere dense set is a set whose closure has empty interior.