# 位相の定義

$X$: set.
$\mathcal{O} \subset P(X)$.

$\mathcal{O}$ is called Topology of $X$ if it fills followings.

1. $\emptyset$, $X \in \mathcal{O}$
2. $(U_i)_{i\in I}$: a family of $\mathcal{O}$. then $\cup_{i\in I} U_i \in \mathcal{O}$
3. $(U_i)_{i\in I}$: a finite family of $\mathcal{O}$. then $\cap_{i\in I} U_i \in \mathcal{O}$