Neighbourhood system

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

Neighbourhood of a point or set

An open neighbourhood of a point (or subset) ${\displaystyle x}$ in a topological space ${\displaystyle X}$ is any open subset ${\displaystyle U}$ of ${\displaystyle X}$ that contains ${\displaystyle x.}$ A neighbourhood of ${\displaystyle x}$ in ${\displaystyle X}$ is any subset ${\displaystyle N\subseteq X}$ that contains some open neighbourhood of ${\displaystyle x}$; explicitly, ${\displaystyle N}$ is a neighbourhood of ${\displaystyle x}$ in ${\displaystyle X}$ if and only if there exists some open subset ${\displaystyle U}$ with ${\displaystyle x\in U\subseteq N}$. Equivalently, a neighborhood of ${\displaystyle x}$ is any set that contains ${\displaystyle x}$ 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) ${\displaystyle x}$ is a filter called the neighbourhood filter for ${\displaystyle x.}$ The neighbourhood filter for a point ${\displaystyle x\in X}$ is the same as the neighbourhood filter of the singleton set ${\displaystyle \{x\}.}$

A neighbourhood basis or local basis (or neighbourhood base or local base) for a point ${\displaystyle x}$ is a filter base of the neighbourhood filter; this means that it is a subset $${\displaystyle {\mathcal {B}}\subseteq {\mathcal {N}}(x)}$$ such that for all ${\displaystyle V\in {\mathcal {N}}(x),}$ there exists some ${\displaystyle B\in {\mathcal {B}}}$ such that ${\displaystyle B\subseteq V.}$ Here, ${\displaystyle {\mathcal {N}}(x)}$ denotes the set of all neighbourhoods of x. That is, for any neighbourhood ${\displaystyle V}$ we can find a neighbourhood ${\displaystyle B}$ in the neighbourhood basis that is contained in ${\displaystyle V.}$

Equivalently, ${\displaystyle {\mathcal {B}}}$ is a local basis at ${\displaystyle x}$ if and only if the neighbourhood filter ${\displaystyle {\mathcal {N}}}$ can be recovered from ${\displaystyle {\mathcal {B}}}$ in the sense that the following equality holds: $${\displaystyle {\mathcal {N}}(x)=\left\{V\subseteq X~:~B\subseteq V{\text{ for some }}B\in {\mathcal {B}}\right\}\!\!\;.}$$ A family ${\displaystyle {\mathcal {B}}\subseteq {\mathcal {N}}(x)}$ is a neighbourhood basis for ${\displaystyle x}$ if and only if ${\displaystyle {\mathcal {B}}}$ is a cofinal subset of ${\displaystyle \left({\mathcal {N}}(x),\supseteq \right)}$ with respect to the partial order ${\displaystyle \supseteq }$ (importantly, this partial order is the superset relation and not the subset relation).