Definition:Finer Topology

Definition
Let $S$ be a set.

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

Let $\tau_1 \supseteq \tau_2$.

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

This can be expressed as:
 * $\tau_1 \ge \tau_2 := \tau_1 \supseteq \tau_2$

Also known as
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.

Also see

 * Definition:Coarser Topology, the opposite of finer topology
 * Definition:Discrete Topology
 * Definition:Indiscrete Topology
 * Discrete Topology is Finest Topology
 * Indiscrete Topology is Coarsest Topology