# Definition:Coarser Topology/Strictly Coarser

## Definition

Let $S$ be a set.

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

Let $\tau_1 \subsetneq \tau_2$.

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

This can be expressed as:

$\tau_1 < \tau_2 := \tau_1 \subsetneq \tau_2$

## Also known as

The terms strictly weaker or strictly smaller are often encountered, meaning the same thing as strictly coarser.

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

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

## Also see

The opposite of strictly coarser is strictly finer.