Neighbourhood system

In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods,[1] or neighbourhood filter ${\mathcal {N}}(x)$ for a point $x$ is the collection of all neighbourhoods of the point $x$ .

Definitions

An open neighbourhood of a subset $S$ of $X$ is any open set $V$ such that $S \subseteq V$ . A neighbourhood of $S$ in $X$ is any subset $T \subseteq X$ such that $T$ contains some open neighborhood of $S$ . Explicitly, this means that $T \subseteq X$ is a neighbourhood of $S$ in $X$ if and only if there is some open set $V$ such that $S \subseteq V \subseteq T$ . The neighbourhood system for any non-empty set $S$ is a filter called the neighbourhood filter for $S$ . The neighbourhood filter for a point $x \in X$ is the same as the neighbourhood filter of the singleton set ${x}.$

A "neighborhood" does not have to be an open set; those neighbourhoods that also happen to be open sets are known as "open neighbourhoods." Similarly, those neighbourhoods that also happen to be closed sets are known as closed neighbourhoods. There are many other types of neighborhoods that are used in Topology and related fields like Functional Analysis. The family of all neighborhoods having a certain "useful" property often forms a neighbourhood basis, although many times, these neighbourhoods are not necessarily open.

Basis

A neighbourhood basis or local basis (or neighbourhood base or local base) for a point $x$ is a filter base of the neighbourhood filter; this means that it is a subset

{\mathcal {B}}(x)\subseteq {\mathcal {N}}(x)

such that for all $V\in {\mathcal {N}}(x)$ , there exists some $B\in {\mathcal {B}}(x)$ such that $B\subseteq V.$ That is, for any neighbourhood $V$ we can find a neighbourhood $B$ in the neighbourhood basis that is contained in $V$ .

Equivalently, ${\mathcal {B}}(x)$ is a local basis at $x$ if and only if the neighbourhood filter ${\mathcal {N}}(x)$ can be recovered from ${\mathcal {B}}(x)$ in the sense that the following equality holds:

{\mathcal {N}}(x)=\left\{V\subseteq X~:~B\subseteq V{\text{ for some }}B\in {\mathcal {B}}(x)\right\}

.[2]

Subbasis

A neighbourhood subbasis at $x$ is a family $𝒮$ of subsets of $X$ , each of which contains $x$ , such that the collection of all possible finite intersections of elements of 𝒮 forms a neighborhood basis at $x$ .

Examples

In any topological space, the neighbourhood system for a point is also a neighbourhood basis for the point.
The set of all open neighborhoods at a point forms a neighbourhood basis at that point.
Given a space $X$ with the indiscrete topology the neighbourhood system for any point $x$ only contains the whole space, ${\mathcal {N}}(x)=\{X\}$
In a metric space, for any point $x$ , the sequence of open balls around $x$ with radius $1/ n$ form a countable neighbourhood basis ${\mathcal {B}}(x)=\{B_{1/n}(x);n\in \mathbb {N} ^{*}\}$ . This means every metric space is first-countable.
In the weak topology on the space of measures on a space $E$ , a neighbourhood base about $\nu$ is given by

\left\{\mu \in {\mathcal {M}}(E):\left|\mu f_{i}-\nu f_{i}\right|<\varepsilon _{i},\,i=1,\ldots ,n\right\}

where

f_{i}

are continuous bounded functions from

E

to the real numbers.

Properties

In a seminormed space, that is a vector space with the topology induced by a seminorm, all neighbourhood systems can be constructed by translation of the neighbourhood system for the origin,

{\mathcal {N}}(x)={\mathcal {N}}(0)+x.

This is because, by assumption, vector addition is separately continuous in the induced topology. Therefore, the topology is determined by its neighbourhood system at the origin. More generally, this remains true whenever the space is a topological group or the topology is defined by a pseudometric.

References

Mendelson, Bert (1990) [1975]. Introduction to Topology (Third ed.). Dover. p. 41. ISBN 0-486-66352-3.
Willard, Stephen (1970). General Topology. Addison-Wesley Publishing. (See Chapter 2, Section 4)

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[1] Mendelson, Bert (1990) [1975]. Introduction to Topology (Third ed.). Dover. p. 41. ISBN 0-486-66352-3.

[2] Willard, Stephen (1970). General Topology. Addison-Wesley Publishing. (See Chapter 2, Section 4)