topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
higher geometry / derived geometry
Ingredients
Concepts
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
Constructions
Examples
derived smooth geometry
Theorems
In topology, a neighbourhood (or neighborhood) of a point in some topological space is a subset 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 .
Let be a topological space and a point. Then:
A subset is a neighbourhood of if there exists an open subset such that and .
A subset is an open neighbourhood of if it is both an open subset and a neighbourhood of ;
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.
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
Last revised on October 3, 2021 at 19:50:00. See the history of this page for a list of all contributions to it.