位相の基底

$(X, \mathcal{O})$: a topological space
$\mathcal{B} \subset \mathcal{O}$

$$ \text{ A basis } \mathcal{B} \text{ for } (X, \mathcal{O}) \Longleftrightarrow {}^{\forall}O \in \mathcal{O} \text{, } {}^{\exists}\mathcal{G} \subset \mathcal{B} \text{ s.t. } \displaystyle O = \cup_{B \in \mathcal{G}} B $$