Definition:Equivalent Topological Bases

Definition
Let $X$ be a set.

Let $\mathbb S_1$ and $\mathbb S_2$ be subsets of $\mathcal P \left({X}\right)$, the power set of $X$.

Let $\mathbb S_1$ and $\mathbb S_2$ be used as a synthetic basis or synthetic sub-basis to generate topologies for $X$.

Let $\vartheta_1$ and $\vartheta_2$ be the topologies arising from $\mathbb S_1$ and $\mathbb S_2$ respectively.

Then $\mathbb S_1$ and $\mathbb S_2$ are equivalent iff $\vartheta_1 = \vartheta_2$.