Definition:Comparable Topologies

Definition

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, by definition of coarser and finer, either:

$\vartheta_1 \subseteq \vartheta_2$

or

$\vartheta_1 \supseteq \vartheta_2$

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$