Definition:Filter Basis

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Definition

Let $S$ be a set.

Let $\mathcal P \left({S}\right)$ be the power set of $S$.

Let $\mathcal B \subset \mathcal P \left({S}\right)$.

Then $\mathcal F := \left\{{V \subseteq X: \exists U \in \mathcal B: U \subseteq V}\right\}$ is a filter on $S$ if and only if the following conditions hold:

$(1): \quad \forall V_1, V_2 \in \mathcal B: \exists U \in \mathcal B: U \subseteq V_1 \cap V_2$
$(2): \quad \varnothing \notin \mathcal B, \mathcal B \ne \varnothing$

Any such $\mathcal B$ is called a filter basis of $\mathcal F$.

$\mathcal F$ is said to be generated by $\mathcal B$ or spanned by $\mathcal B$.

This is proved in Filter Basis Generates Filter.

Equivalent Filter Bases

Two filter bases are equivalent if and only if they both generate the same filter.

Also known as

A filter basis is also known as a filter base.

Linguistic Note

The plural of basis is bases.

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