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) $$\forall V_1, V_2 \in \mathcal B: \exists U \in \mathcal B: U \subseteq V_1 \cap V_2$$
 * 2) $$\varnothing \not \in \mathcal B, \mathcal B \ne \varnothing$$

Any such $$\mathcal B$$ is called a filter basis (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