Definition:Coarser Topology

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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:

$\vartheta_1 \le \vartheta_2 := \vartheta_1 \subseteq \vartheta_2$

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 $\vartheta_1 < \vartheta_2$

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.

The opposite of coarser is finer.