topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory




In topology, a neighbourhood (or neighborhood) of a point xx in some topological 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.


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

  1. A subset UXU \subset X is a neighbourhood of xx if there exists an open subset OXO \subset X such that xOx \in O and OUO \subset U.

  2. A subset UXU \subset X is an open neighbourhood of xx if it is both an open subset and a neighbouhood of xx;

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

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


  • 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\subsetformal = arbitrary order infinitesimal\subsetlocal = stalkwise\subsetfinite
\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 geometry𝔽 p\mathbb{F}_p finite field p\mathbb{Z}_p p-adic integers (p)\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

Revised on May 15, 2017 03:59:56 by Urs Schreiber (