Definition:Finer Topology/Strictly Finer

Definition

Let $S$ be a set.

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

Let $\tau_1 \supsetneq \tau_2$.

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

This can be expressed as:

$\tau_1 > \tau_2 := \tau_1 \supsetneq \tau_2$

Also known as

The terms strictly stronger or strictly larger are often encountered, meaning the same thing as strictly finer.

Unfortunately, the term strictly weaker is also sometimes encountered, meaning exactly the same thing.

To remove any ambiguity as to which one is meant, it is recommended that strictly finer be used exclusively.

Also see

The opposite of strictly finer is strictly coarser.

Sources

• 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Chapter $\text {I}$: Topological Spaces: $1$. Open Sets and Closed Sets