近傍系による連続性の表現

$(X, \mathcal{O}_X)$, $(Y, \mathcal{O}_Y)$: topological space
$f : X \longmapsto Y$: map
$\mathcal{N}_X(a)$ : neighbourhood system
$\mathcal{N}_Y(a)$ : neighbourhood system

$$ {}^{\forall}x \in X \text{, } {}^{\forall}U \in \mathcal{N}_Y(f(x)) \text{, } {}^{\exists}V \in \mathcal{N}_X(x) \text{, } f(V) \subset U \Longleftrightarrow f \text { is continuous } $$