Definition:Equivalent Topological Bases

Definition

Let $X$ be a set.

Let $\mathbb S_1$ and $\mathbb S_2$ be subsets of $\powerset X$, 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 $\tau_1 = \tau_2$.

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction