Definition:Expansion of Topology

Definition
Let $X$ be a set.

Let $\vartheta_1$ and $\vartheta_2$ be topologies on $X$ such that $\vartheta_1 \subseteq \vartheta_2$.

Then $\vartheta_2$ is an expansion of $\vartheta_1$.

Also see
By definition, it can be seen that $\vartheta_2$ is an expansion of $\vartheta_1$ iff $\vartheta_1$ is coarser than $\vartheta_2$.

Alternatively, iff $\vartheta_2$ is finer than $\vartheta_1$.