#topology

## Definition
A *topology* of/on a set $X$ is a collection $\topology{O} \subseteq \topology{P}(X)$ of subsets such that
 - $\emptyset, X \in \mathcal{O}$.
 - If $O_1, \dots, O_n \in \mathcal{O}$, then $\bigcap_{j = 1}^n O_j \in \mathcal{O}$ as well.
 - If $O_1, O_2, \dots \in \mathcal{O}$, then $\bigcup_{j = 1}^\infty O_j \in \mathcal{O}$ as well.

We call $(X, \mathcal{O})$ a *topological space*.