Power Set as Complete Lattice (冪集合の完備束) (Section I.3)
- Power Set as Complete Lattice (冪集合の完備束) (Section I.3) #Card
- $(2^X, \subset)$ is a complete lattice for every set $X$, with $\bigvee \mathcal{A} = \bigcup \mathcal{A}$ and $\bigwedge \mathcal{A} = \bigcap \mathcal{A}$.
For each set $X$, the set $L = 2^X$ of subsets of $X$ is a complete lattice with respect to inclusion $\subset$: for any $\mathcal{A} \subset 2^X$, $$\bigvee \mathcal{A} = \bigcup_{A \in \mathcal{A}} A, \qquad \bigwedge \mathcal{A} = \bigcap_{A \in \mathcal{A}} A.$$
- フィルターの束 $(\overline{\mathbb{F}}X, \subset)$（cards/topology/prop-ii-3-2）や収束の束（cards/topology/prop-iii-3-7）の原型。
- 有限集合上の束は必然的に完備（Example I.3.1 の $2^{{0,1}}$ など）。
Set X の CompleteLattice（sSup_eq_sUnion 相当; Set.sUnion, Set.sInter）