Filter

Remarks

Some authors, and Mathlib, drop clause 2. Then the only family that satisfies clauses 1, 3 and 4 and contains $\emptyset$ is the power set of $X$. A filter in the sense above is then called a proper filter.

Example: for a point $x$ of a topological space $X$, the neighbourhoods of $x$ form a filter $\mathcal{N}(x)$, the neighbourhood filter of $x$. Clause 2 holds because each neighbourhood contains $x$, and clause 4 holds because the intersection of two neighbourhoods is a neighbourhood.

Example: on an infinite set $X$, the sets with finite complement form a filter, the Fréchet filter. Clause 2 holds because $X$ is infinite, and clause 4 holds because the union of two finite sets is finite.

Example: for $a \in X$, the family $\set{A \subseteq X : a \in A}$ is a filter, the principal filter at $a$.

Non-example: on $\N$, the family of sets that contain $0$ or contain $1$ satisfies clauses 1, 2 and 3. It fails clause 4: it contains $\set{0}$ and $\set{1}$, but not $\set{0} \cap \set{1} = \emptyset$.

A nonempty family $\mathcal{B}$ of nonempty subsets of $X$ is a filter base if the intersection of any two members contains a member. The supersets of members of a filter base form a filter.