Definition 1

A filter on a set $X$ is a set $\mathfrak{F}$ of subsets of $X$ which has the following properties:

$(\mathrm{F}_1)$ Every subset of $X$ which contains a set of $\mathfrak{F}$ belongs to $\mathfrak{F}$. $(\mathrm{F}_\mathrm{II})$ Every finite intersection of sets of $\mathfrak{F}$ belongs to $\mathfrak{F}$. $(\mathrm{F}_\mathrm{III})$ The empty set is not in $\mathfrak{F}$.

It follows from $(\mathrm{F}_\mathrm{II})$ and $(\mathrm{F}_\mathrm{III})$ that every finite intersection of sets of $\mathfrak{F}$ is non-empty.

A filter $\mathfrak{F}$ on $X$ defines a structure on $X$, the axioms of which are $(\mathrm{F}_1)$, $(\mathrm{F}_\mathrm{II})$ and $(\mathrm{F}_\mathrm{III})$; this structure is called a structure of a filtered set, and the set $X$ endowed with this structure is called a set filtered by $\mathfrak{F}$.

Axiom $(\mathrm{F}_\mathrm{II})$ is equivalent to the conjunction of the following two axioms: $(\mathrm{F}_{\mathrm{II}\,\mathrm{a}})$ The intersection of two sets of $\mathfrak{F}$ belongs to $\mathfrak{F}$. $(\mathrm{F}_{\mathrm{II}\,\mathrm{b}})$ $X$ belongs to $\mathfrak{F}$.

Axioms $(\mathrm{F}_{\mathrm{II}\,\mathrm{b}})$ and $(\mathrm{F}_\mathrm{III})$ show that there is no filter on the empty set.

In order for a set of subsets which satisfies $(\mathrm{F}_1)$ also to satisfy $(\mathrm{F}_{\mathrm{II}\,\mathrm{b}})$ it is necessary and sufficient that it is not empty. A set of subsets which satisfies $(\mathrm{F}_1)$ also satisfies $(\mathrm{F}_\mathrm{III})$ if and only if it is different from $\mathfrak{P}(X)$.

Examples of filters. 1) If $X \neq \varnothing$, the set of subsets consisting of $X$ alone is a filter on $X$. More generally, the set of all subsets of $X$ which contain a given non-empty subset $A$ of $X$ is a filter on $X$.

2) In a topological space $X$, the set of all neighbourhoods of an arbitrary non-empty subset $A$ of $X$ (and in particular the set of all neighbourhoods of a point of $X$) is a filter, called the neighbourhood filter of $A$.

3) If $X$ is an infinite set, the complements of the finite subsets of $X$ are the elements of a filter. The filter of complements of finite subsets of the set $\mathbf{N}$ of integers $\geqslant 0$ is called the Fréchet filter.