Definition:Comparable Topologies

Theorem
Let $S$ be a set.

Let $\vartheta_1$ and $\vartheta_2$ be topologies on $S$.

Then $\vartheta_1$ and $\vartheta_2$ are comparable iff either:
 * $\vartheta_1$ is coarser than $\vartheta_2$

or


 * $\vartheta_1$ is finer than $\vartheta_2$

That is, either:
 * $\vartheta_1 \subseteq \vartheta_2$

or


 * $\vartheta_1 \supseteq \vartheta_2$