Order on Families of Sets (集合族の粗細) (Section IV.1)
- Order on Families of Sets (集合族の粗細) (Section IV.1) #Card
- $\mathcal{A}$ is coarser than $\mathcal{D}$ ($\mathcal{D}$ finer than $\mathcal{A}$), written $\mathcal{A} \leq \mathcal{D}$, if for each $A \in \mathcal{A}$ there is $D \in \mathcal{D}$ such that $D \subset A$.
If $\mathcal{A}$ and $\mathcal{D}$ are families of subsets of $X$, then $\mathcal{A}$ is said to be coarser than $\mathcal{D}$ (equivalently, $\mathcal{D}$ is finer than $\mathcal{A}$), in symbols $\mathcal{A} \leq \mathcal{D}$, if for each $A \in \mathcal{A}$, there is $D \in \mathcal{D}$ such that $D \subset A$.
- $\mathcal{A} \subset \mathcal{D} \implies \mathcal{A} \leq \mathcal{D}$；$\mathcal{D}$ が isotone なら $\mathcal{A} \leq \mathcal{D} \implies \mathcal{A} \subset \mathcal{D}$。
- $\mathcal{B}$ is a base of an isotone family $\mathcal{A}$ if $\mathcal{B} \subset \mathcal{A}$ and $\mathcal{B} \geq \mathcal{A}$（cards/topology/filter-base の一般化）。
フィルター基底間の対応は Filter.HasBasis / FilterBasis