Comparison of topologies

Definition

Let $X$ be a set and let $\tau_1$ and $\tau_2$ be topologies on $X$. If $\tau_1 \subseteq \tau_2$, then $\tau_1$ is coarser (or weaker) than $\tau_2$, and $\tau_2$ is finer (or stronger) than $\tau_1$. If also $\tau_1 \neq \tau_2$, then $\tau_1$ is strictly coarser and $\tau_2$ is strictly finer.

Remarks

On each set $X$, the indiscrete topology is the coarsest topology and the discrete topology is the finest topology. We have $\tau_1 \subseteq \tau_2$ if and only if the identity map $(X, \tau_2) \to (X, \tau_1)$ is continuous. Two topologies need not be comparable: on $\{a, b\}$, the topologies $\{\varnothing, \{a\}, X\}$ and $\{\varnothing, \{b\}, X\}$ are not comparable.

Explanations

L1 human

A finer topology has more open sets, so it can separate points in more ways. A coarser topology has fewer open sets, so more sequences converge and more maps into it are continuous.