CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
A neighbourhood (or neighborhood) of a point $x$ in some space $X$ is a set $U$ 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 $x$.
In a topological space $X$, let a point in $X$ be an element of $X$ and let a set in $X$ be a subset of $X$.
Then a set $U$ is a neighbourhood of a point $x$ if there exists an open set $G$ such that $x \in G$ and $G \subseteq U$.
A set $U$ is an open neighbourhood of a point $x$ if $U$ is open and $x \in U$; 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.
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
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 |