Editing Glossary of general topology (section)

== N ==

; Nearly open: see ''preopen''. See also: [[almost open map]].

;[[Neighbourhood (mathematics)|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 (mathematics)|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.)

;[[Local base|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 (mathematics)|Net]]: A [[net (mathematics)|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''<sub>α</sub>), where α is an [[index set|index variable]] ranging over ''A''. Every [[sequence]] is a net, taking ''A'' to be the directed set of [[natural number]]s with the usual ordering.

;[[Normal space|Normal]]: A space is [[normal space|normal]] if any two disjoint closed sets have disjoint neighbourhoods.<ref name=ss162/>  Every normal space admits a [[partition of unity]].

;[[T4 space|Normal Hausdorff]]: A [[T4 space|normal Hausdorff]] space (or [[T4 space|'''T<sub>4</sub>''' space]]) is a normal T<sub>1</sub> space. (A normal space is Hausdorff [[if and only if]] it is T<sub>1</sub>, so the terminology is consistent.) Every normal Hausdorff space is Tychonoff.

;[[Nowhere dense set|Nowhere dense]]: A [[nowhere dense set]] is a set whose closure has empty interior.