Definition:Filter Basis/Definition 1
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Definition
Let $S$ be a set.
Let $\powerset S$ be the power set of $S$.
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 $\FF$.
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.
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