Based on this setup and definitions, I need to solve this.
How are the three topologies on X related to each other (in terms of coarser and finer)?
Definition of coarser and finer topology: Let $X$ be a set and $T_1$ and $T_2$ be topologies on $X$. If $T_1$ $\subset T_2$, we say that $T_1$ is coarser than $T_2$ and that $T_2$ is finer than $T_1$. If $T_1$ $\subset$ $T_2$ and $T_1$ $\neq$ $T_2$, we say that $T_1$ is strictly coarser than $T_2$ and that $T_2$ is strictly finer than $T_1$.
Taking a look at the definitions where we have 3 topologies, we can see that $T_s$ is the subspace topology that is inherited from the product topology,$T_r$ is the topology by railroad metric and $T_c$ is the coherent topology. I think it should be $T_s \subset T_c$ but not sure whether $T_s \subset T_r$. The other direction should be similar to one direction.
Need help in this. Appreciate your help & support.