Topological minor and independence number

Question

If $G$ and $X$ are graphs with $G=TX$, ($X$ is a topological minor of $G$) is there any sort of relation between $\alpha(G)$ and $\alpha(X)$ (the independence numbers of both). In addition if finding the largest independent set in $G$ is NP-hard is it always the case that finding it in $X$ will also be NP-hard? I suspect an answer to my second question may provide an answer to my first. I know that the smallest vertex cover problem and maximum independent set problem are related so I also want to extend my question to vertex covers, is there a relationship between the smallest vertex cover of $G$ and the smallest vertex cover of $X$?