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 $$\vartheta_1 = \vartheta_2$$.