Neighbourhood (mathematics)

A neighbourhood of a point is a set containing an open set that contains that point.

An open neighbourhood of a point $x$ in a topological space $(X,\mathcal{T})$ is an open set $U$ containing $x$. A general neighbourhood need only contain an open neighbourhood; it need not itself be open. A finite intersection of open neighbourhoods of the same point is again an open neighbourhood, by the topology axioms.

The neighbourhood system at a point consists of all its neighbourhoods. A neighbourhood basis is a subfamily cofinal under reverse inclusion.

A neighbourhood basis at a point is a family of neighbourhoods such that every neighbourhood of the point contains one of them.