derived smooth geometry
A neighbourhood (or neighborhood) of a point in some space is a set such that there is enough room around in to move in any direction (but perhaps not very far). One writes , , or any of the six other obvious variations to indicate that is a neighbourhood of .
Then a set is a neighbourhood of a point if there exists an open set such that and .
A set is an open neighbourhood of a point if is open and ; 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.
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 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
|geometry||point||first order infinitesimal||formal = arbitrary order infinitesimal||local = stalkwise||finite|
|smooth functions||derivative||Taylor series||germ||smooth function|
|curve (path)||tangent vector||jet||germ of curve||curve|
|smooth space||infinitesimal neighbourhood||formal neighbourhood||open neighbourhood|
|function algebra||square-0 ring extension||nilpotent ring extension/formal completion||ring extension|
|arithmetic geometry||finite field||p-adic integers||localization at (p)||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|