Definition:Coarser Topology

Definition
Let $S$ be a set.

Let $\tau_1$ and $\tau_2$ be topologies on $S$.

Let $\tau_1 \subseteq \tau_2$.

Then $\tau_1$ is said to be coarser than $\tau_2$.

This can be expressed as:
 * $\tau_1 \le \tau_2 := \tau_1 \subseteq \tau_2$

Also known as
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.

Also see

 * Definition:Finer Topology, the opposite of coarser topology.
 * Definition:Discrete Topology
 * Definition:Indiscrete Topology
 * Discrete Topology is Finest Topology
 * Indiscrete Topology is Coarsest Topology