# Definition:Finer Topology

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## Definition

Let $S$ be a set.

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

### Definition 1

$\tau_1$ is **finer** than $\tau_2$ if and only if $\tau_1 \supseteq \tau_2$.

### Definition 2

$\tau_1$ is **finer** than $\tau_2$ if and only if the identity mapping $(S, \tau_1) \to (S, \tau_2)$ is continuous.

This can be expressed as:

- $\tau_1 \ge \tau_2 := \tau_1 \supseteq \tau_2$

### Strictly Finer

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 **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.