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# Contents

## 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.

Examples of sequences of local structures

geometrypointfirst order infinitesimal$\subset$formal = arbitrary order infinitesimal$\subset$local = stalkwise$\subset$finite
$\leftarrow$ differentiationintegration $\to$
smooth functionsderivativeTaylor seriesgermsmooth function
curve (path)tangent vectorjetgerm of curvecurve
smooth spaceinfinitesimal neighbourhoodformal neighbourhoodgerm of a spaceopen neighbourhood
function algebrasquare-0 ring extensionnilpotent ring extension/formal completionring extension
arithmetic geometry$\mathbb{F}_p$ finite field$\mathbb{Z}_p$ p-adic integers$\mathbb{Z}_{(p)}$ localization at (p)$\mathbb{Z}$ integers
Lie theoryLie algebraformal grouplocal Lie groupLie group
symplectic geometryPoisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

Last revised on October 3, 2021 at 19:50:00. See the history of this page for a list of all contributions to it.