Definition:Finer Topology

Let $$S$$ be a set.

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

Let $$\vartheta_1 \supseteq \vartheta_2$$.

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

Strictly Finer
As above, but let $$\vartheta_1 \supset \vartheta_2$$, that is, $$\vartheta_1 \supseteq \vartheta_2$$ but $$\vartheta_1 \ne \vartheta_2$$.

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

Coarser
The opposite of finer is coarser.

Stronger, Weaker
The term stronger is often encountered, meaning the same thing as finer.

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

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