Definition:Equivalent Topological Bases

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Let $X$ be a set.

Let $\mathbb S_1$ and $\mathbb S_2$ be subsets of $\mathcal P \paren{X}$, 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 $\tau_1$ and $\tau_2$ be the topologies arising from $\mathbb S_1$ and $\mathbb S_2$ respectively.

Then $\mathbb S_1$ and $\mathbb S_2$ are equivalent if and only if $\tau_1 = \tau_2$.