Let $X$ be a D11: Set such that

(i)	$\mathcal{P}(X)$ is the D80: Power set of $X$

A D11: Set $\mathcal{T} \subseteq \mathcal{P}(X)$ is a topology on $X$ if and only if

(1)	\begin{equation} \emptyset, X \in \mathcal{T} \end{equation}
(2)	\begin{equation} \forall \, \mathcal{S} \subseteq \mathcal{T} : \cup \mathcal{S} \in \mathcal{T} \end{equation}
(3)	\begin{equation} \forall \, N \in 1, 2, 3, \ldots : \forall \, E_1, \dots, E_N \in \mathcal{T} : \bigcap_{n = 1}^N E_n \in \mathcal{T} \end{equation}

Let $X$ be a D11: Set such that

(i)	$\mathcal{P}(X)$ is the D80: Power set of $X$

A D11: Set $\mathcal{T} \subseteq \mathcal{P}(X)$ is a topology on $X$ if and only if

(1)	\begin{equation} \emptyset, X \in \mathcal{T} \end{equation}
(2)	\begin{equation} \forall \, \mathcal{S} \subseteq \mathcal{T} : \bigcup_{S \in \mathcal{S}} S \in \mathcal{T} \end{equation}
(3)	\begin{equation} \forall \, N \in 1, 2, 3, \ldots : \forall \, E_1, \dots, E_N \in \mathcal{T} : \bigcap_{n = 1}^N E_n \in \mathcal{T} \end{equation}