Definition:Filter Basis

Definition

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

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

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

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

$(2): \quad \varnothing \not \in \mathcal B, \mathcal B \ne \varnothing$

Any such $\mathcal B$ is called a filter base or filter basis of $\mathcal F$ (plural: filter bases).

$\mathcal F$ 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 iff they both generate the same filter.

Also see

• Basis (Topology)

• Filter Sub-Basis

Sources

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