Definition:Coarser Topology

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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:

$\tau_1 \le \tau_2 := \tau_1 \subseteq \tau_2$


Strictly Coarser

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 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


Sources