Definition:Filter Basis/Definition 2

Definition

Let $S$ be a set.

Let $\BB$ be a subset of a filter $\FF$ on $S$ such that $\BB \ne \O$.

Then $\BB$ is a filter basis of $\FF$ if and only if:

$\forall U \in \FF: \exists V \in \BB: V \subseteq U$

Generated Filter

$\FF$ is said to be generated by $\BB$.

Also known as

A filter basis is also known as a filter base.

Also see

• Equivalence of Definitions of Filter Basis

Linguistic Note

The plural of basis is bases.

This is properly pronounced bay-seez, not bay-siz.

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Filters