Definition:Basis (Topology)/Synthetic Basis/Definition 2

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Definition

Let $S$ be a set.


A synthetic basis on $S$ is a subset $\mathcal B \subseteq \mathcal P \left({S}\right)$ of the power set of $S$ such that:

$\mathcal B$ is a cover for $S$
$\forall U, V \in \mathcal B: \forall x \in U \cap V: \exists W \in \mathcal B: x \in W \subseteq U \cap V$