Definition:Coarser Topology

Let $$S$$ be a set.

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

Let $$\vartheta_1 \subseteq \vartheta_2$$.

Then $$\vartheta_1$$ is said to be coarser than $$\vartheta_2$$.

Strictly Coarser
As above, but let $$\vartheta_1 \subset \vartheta_2$$, that is, $$\vartheta_1 \subseteq \vartheta_2$$ but $$\vartheta_1 \ne \vartheta_2$$.

Then $$\vartheta_1$$ is said to be strictly coarser than $$\vartheta_2$$.

Finer
The opposite of coarser is finer.

Weaker, Stronger
The term weaker is often encountered, meaning the same thing as coarser.

Unfortunately, the term stronger is also sometimes encountered, meaning exactly the same thing.

To remove any ambiguity as to which one is meant, it is recommended that coarser be used exclusively.