neighborhood in nLab

Contents

Contents

Idea

In topology, a neighbourhood (or neighborhood) of a point xx in some topological space XX is a subset UU such that there is enough room around xx in UU to move in any direction (but perhaps not very far). One writes x∈U ∘x \in U^\circ, U∋∘xU \stackrel{\circ}\ni x, or any of the six other obvious variations to indicate that UU is a neighbourhood of xx.

Definitions

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

1. A subset U⊂XU \subset X is a neighbourhood of xx if there exists an open subset O⊂XO \subset X such that x∈Ox \in O and O⊂UO \subset U.

2. A subset U⊂XU \subset X is an open neighbourhood of xx if it is both an open subset and a neighbourhood of xx;

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.

• neighbourhood base