Find two different topologies on $\displaystyle \mathbb{R}$ where one is strictly finer, yet the two are homeomorphic
$\displaystyle (Z, T_1)$ and $\displaystyle (Q, T_2)$ as subspaces of R with a discrete topology.
Let $\displaystyle B_1$ and $\displaystyle B_2$ be bases for topologies $\displaystyle T_1$ and $\displaystyle T_2$.
$\displaystyle T_2$ is strictly finer than $\displaystyle T_1$, since for each x in Z and for each basis element $\displaystyle \{x\} \in B_1$ containing x, we have a basis element $\displaystyle \{x\}$ in $\displaystyle B_2$ that contains x. The converse is not necessarily true.
Since there is a bijection between Z and Q, we have a homeomorphism between $\displaystyle (Z, T_1)$ and $\displaystyle (Q, T_2)$ as subspaces of R with a discrete topology.