Definition:Finer Topology

Definition
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 $$\vartheta_2$$.

This can be expressed as:
 * $$\vartheta_1 \ge \vartheta_2 \ \stackrel {\mathbf {def}} {=\!=} \ \vartheta_1 \supseteq \vartheta_2$$

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.

Alternative names
The terms stronger or larger are 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.