Definition:Comparable Topologies
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Definition
Let $S$ be a set.
Let $\tau_1$ and $\tau_2$ be topologies on $S$.
Then $\tau_1$ and $\tau_2$ are comparable if and only if either:
- $\tau_1$ is coarser than $\tau_2$
or
- $\tau_1$ is finer than $\tau_2$
That is, by definition of coarser and finer, either:
- $\tau_1 \subseteq \tau_2$
or
- $\tau_1 \supseteq \tau_2$
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