Definition:Expansion of Topology

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Let $S$ be a set.

Let $\tau_1$ and $\tau_2$ be topologies on $S$ such that $\tau_1 \subseteq \tau_2$.

Then $\tau_2$ is an expansion of $\tau_1$.

Also see

By definition, it can be seen that $\tau_2$ is an expansion of $\tau_1$ if and only if $\tau_1$ is coarser than $\tau_2$.

That is, if and only if $\tau_2$ is finer than $\tau_1$.