# Definition:Finer Topology

From ProofWiki

## Definition

Let $S$ be a set.

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

Let $\tau_1 \supseteq \tau_2$.

Then $\tau_1$ is said to be **finer** than $\tau_2$.

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.

## Also see

- Definition:Coarser Topology, the opposite of
**finer topology** - Definition:Discrete Topology
- Definition:Indiscrete Topology
- Discrete Topology is Finest Topology
- Indiscrete Topology is Coarsest Topology

## Sources

- 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): $\S 1.1$ - 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 1$ - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $3.1$: Topological Spaces