Definition:Filter on a Set

Definition

Let $X$ be a set, and $\mathcal P \left({X}\right)$ be the power set of $X$.

A filter on $X$ (or filter of $X$) is a set $\mathcal F \subset \mathcal P \left({X}\right)$ which satisfies the following conditions:

$(1): \quad X \in \mathcal F$

$(2): \quad \varnothing \notin \mathcal F$

$(3): \quad U, V \in \mathcal F \implies U \cap V \in \mathcal F$

$(4): \quad \forall U \in \mathcal F: U \subseteq V \subseteq X \implies V \in \mathcal F$

Filtered Set

The set $X$ on which the filter has been applied is called a set filtered by $\mathcal F$, or just filtered set.

Trivial Filter

A filter $\mathcal F$ on a set $X$ as defined above specifically does not include the empty set $\varnothing$.

If a filter $\mathcal F$ were to include $\varnothing$, then from Empty Set Subset of All it would follow that every subset of $X$ would have to be in $\mathcal F$, and so $\mathcal F = \mathcal P \left({X}\right)$.

Such a filter is called the trivial filter on $X$.

Finite Intersection

It follows directly by Principle of Mathematical Induction from:

$U, V \in \mathcal F \implies U \cap V \in \mathcal F$

that the intersection of any finite number of sets of $\mathcal F$ is also an element of $\mathcal F$.

Some treatments of this subject start with this as an axiom.

Also see

• See Filter on a Set is a Proper Filter.

• Results about filters can be found here.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$: Filters