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Definition D86
Topology

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

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) $$\emptyset, X \in \mathcal{T}$$ (2) $$\forall \, \mathcal{S} \subseteq \mathcal{T} : \bigcup_{S \in \mathcal{S}} S \in \mathcal{T}$$ (3) $$\forall \, N \in 1, 2, 3, \ldots : \forall \, E_1, \dots, E_N \in \mathcal{T} : \bigcap_{n = 1}^N E_n \in \mathcal{T}$$
Children
 ▶ Bottom topology ▶ Pushforward topology ▶ Set of topologies ▶ Subtopology