nLab
neighborhood

Context

Topology

Higher geometry

Contents

Idea

A neighbourhood (or neighborhood) of a point xx in some space XX is a set UU such that there is enough room around xx in UU to move in any direction (but perhaps not very far). One writes xU x \in U^\circ, UxU \stackrel{\circ}\ni x, or any of the six other obvious variations to indicate that UU is a neighbourhood of xx.

Definitions

In a topological space XX, let a point in XX be an element of XX and let a set in XX be a subset of XX.

Then a set UU is a neighbourhood of a point xx if there exists an open set GG such that xGx \in G and GUG \subseteq U.

A set UU is an open neighbourhood of a point xx if UU is open and xUx \in U; many authors use the simple term ‘neighbourhood’ only for open neibhourhoods.

As the term implies, an open neighbourhood is precisely a neighbourhood that is open. One can also define closed neighbourhoods, compact neighbourhoods, etc.

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.

Examples of sequences of infinitesimal and local structures

geometryfirst order infinitesimal\subsetformal = arbitrary order infinitesimal\subsetlocal = stalkwise\subsetfinite
\leftarrow differentiationintegration \to
smooth functionsderivativeTaylor seriesgermsmooth function
curve (path)tangent vectorjetgerm of curvecurve
smooth spaceinfinitesimal neighbourhoodformal neighbourhoodopen neighbourhood
function algebrasquare-0 ring extensionnilpotent ring extension/formal completionring extension
arithmetic geometryp-adic integersintegers
Lie theoryLie algebraformal grouplocal Lie groupLie group
symplectic geometryPoisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

Revised on July 23, 2014 03:41:20 by Urs Schreiber (89.204.154.176)