Definition:Finer Topology

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.

Also see

• Equivalence of Definitions of Finer Topology
• Definition:Coarser Topology, the opposite of finer topology
• Definition:Discrete Topology
• Definition:Indiscrete Topology
• Discrete Topology is Finest Topology
• Indiscrete Topology is Coarsest Topology