Definition:Coarser Topology

Definition
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 $$\vartheta_2$$.

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

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 $$\vartheta_2$$, and we can write $$\vartheta_1 < \vartheta_2$$

Finer
The opposite of coarser is finer.

Alternative names
The terms weaker or smaller are 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.