nLab neighborhood

Idea

In topology, a neighbourhood (or neighborhood) of a point $x$ in some topological space $X$ is a subset $U$ such that there is enough room around $x$ in $U$ to move in any direction (but perhaps not very far). One writes $x \in U^\circ$, $U \stackrel{\circ}\ni x$, or any of the six other obvious variations to indicate that $U$ is a neighbourhood of $x$.

Definitions

Let $(X,\tau)$ be a topological space and $x \in X$ a point. Then:

1. A subset $U \subset X$ is a neighbourhood of $x$ if there exists an open subset $O \subset X$ such that $x \in O$ and $O \subset U$.

2. A subset $U \subset X$ is an open neighbourhood of $x$ if it is both an open subset and a neighbourhood of $x$;

Beware, some authors use “neighbourhood” as a synonym for “open neighbourhood”.

Similarly one says that a closed neighbourhood or compact neighbourhood etc. is a neighbourhood that is also a closed subset or compact subspace, respectively.

Properties

• When definitions of topological concepts are given in terms of neighbourhoods, it often makes no difference if the neighbourhoods are required to be open or not. There should be some deep logical reason for this ….

• The neighbourhoods of a given point form a proper filter, the neighbourhood filter of that point. A local (sub)base for the topology at that point is a (sub)base for that filter.

• The concept of topological space can be defined by taking the neighbourhood relation as primitive. One axiom is more complicated than the others; if it is dropped, then the result is the definition of pretopological space.

Related concepts

• neighbourhood base

Examples of sequences of local structures

geometry	point	first order infinitesimal	$\subset$	formal = arbitrary order infinitesimal	$\subset$	local = stalkwise	$\subset$	finite
			$\leftarrow$ differentiation		integration $\to$
smooth functions		derivative		Taylor series		germ		smooth function
curve (path)		tangent vector		jet		germ of curve		curve
smooth space		infinitesimal neighbourhood		formal neighbourhood		germ of a space		open neighbourhood
function algebra		square-0 ring extension		nilpotent ring extension/formal completion				ring extension
arithmetic geometry	$\mathbb{F}_p$ finite field			$\mathbb{Z}_p$ p-adic integers		$\mathbb{Z}_{(p)}$ localization at (p)		$\mathbb{Z}$ integers
Lie theory		Lie algebra		formal group		local Lie group		Lie group
symplectic geometry		Poisson manifold		formal deformation quantization		local strict deformation quantization		strict deformation quantization