Finer topology ($\tau \subseteq \sigma$)

For two topologies on the same set $X$, $\sigma$ is finer than $\tau$ when every $\tau$-open set is $\sigma$-open. Equivalently, the identity function $(X,\sigma )\to (X,\tau )$ is continuous, by the definition using preimages of open sets. The inclusion may be an equality; strictly finer means proper inclusion.