Definition:Basis (Topology)/Synthetic Basis/Definition 1

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Definition

Let $S$ be a set.


A synthetic basis on $S$ is a subset $\BB \subseteq \powerset S$ of the power set of $S$ such that:

\((\text B 1)\)   $:$   $\BB$ is a cover for $S$      
\((\text B 2)\)   $:$     \(\ds \forall U, V \in \BB:\) $\exists \AA \subseteq \BB: U \cap V = \bigcup \AA$      

That is, the intersection of any pair of elements of $\BB$ is a union of sets of $\BB$.


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