Neighbourhood system

In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter

N

(

x

)

{\displaystyle {\mathcal {N}}(x)}

for a point

x

{\displaystyle x}

in a topological space is the collection of all neighbourhoods of

x

.

{\displaystyle x.}

Definitions

Neighbourhood of a point or set

An open neighbourhood of a point (or subset)

x

{\displaystyle x}

in a topological space

X

{\displaystyle X}

is any open subset

U

{\displaystyle U}

of

X

{\displaystyle X}

that contains

x

.

{\displaystyle x.}

A neighbourhood of

x

{\displaystyle x}

in

X

{\displaystyle X}

is any subset

N

⊆

X

{\displaystyle N\subseteq X}

that contains some open neighbourhood of

x

{\displaystyle x}

;

explicitly,

N

{\displaystyle N}

is a neighbourhood of

x

{\displaystyle x}

in

X

{\displaystyle X}

if and only if there exists some open subset

U

{\displaystyle U}

with

x

∈

U

⊆

N

{\displaystyle x\in U\subseteq N}

.

Equivalently, a neighborhood of

x

{\displaystyle x}

is any set that contains

x

{\displaystyle x}

in its topological interior.

Importantly, a "neighbourhood" does not have to be an open set; those neighbourhoods that also happen to be open sets are known as "open neighbourhoods."

Similarly, a neighbourhood that is also a closed (respectively, compact, connected, etc.) set is called a closed neighbourhood (respectively, compact neighbourhood, connected neighbourhood, etc.).

There are many other types of neighbourhoods that are used in topology and related fields like functional analysis.

The family of all neighbourhoods having a certain "useful" property often forms a neighbourhood basis, although many times, these neighbourhoods are not necessarily open. Locally compact spaces, for example, are those spaces that, at every point, have a neighbourhood basis consisting entirely of compact sets.

Neighbourhood filter

The neighbourhood system for a point (or non-empty subset)

x

{\displaystyle x}

is a filter called the neighbourhood filter for

x

.

{\displaystyle x.}

The neighbourhood filter for a point

x

∈

X

{\displaystyle x\in X}

is the same as the neighbourhood filter of the singleton set

{

x

}

.