# Definition:Coarser Topology/Strictly Coarser

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## Contents

## 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.

## Sources

- 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): $\S 1.1$