Definition:Filter Basis
(Redirected from Definition:Equivalent Filter Bases)
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Definition
Let $S$ be a set.
Let $\powerset S$ denote the power set of $S$.
Definition 1
Let $\BB \subset \powerset S$ such that $\O \notin \BB$ and $\BB \ne \O$.
Then $\FF := \set {V \subseteq S: \exists U \in \BB: U \subseteq V}$ is a filter on $S$ if and only if:
- $\forall V_1, V_2 \in \BB: \exists U \in \BB: U \subseteq V_1 \cap V_2$
Such a $\BB$ is called a filter basis of $\FF$.
Definition 2
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
Linguistic Note
The plural of basis is bases.
This is properly pronounced bay-seez, not bay-siz.