As I learned in the beginning of Topology, if I have two topologies $\tau_1, \tau_2$ on a space $X$ such that $$ \tau_1 \subseteq \tau_2 $$ then $\tau_2$ is called finer then $\tau_1$ and $\tau_1$ is called coarser then $\tau_2$. Roughly speaking "finer" means has more elements, and "coarser" means has less elements (which makes intuitively sense cause the notion of "closeness of points" axiomatizied by a topology is finer).
Now I read about open covers and refinements. The definitions are here. Now a cover is finer if it actually contains less open sets, and coarser if it contains more, am I right? So here the meaning of finer is totatlly different? (maybe intuitively finer means for a point it could be more accurately said in which open set it falls, and if there are more the more "fuzzy" the "location" of the point is). Am I right, and maybe is there some connections between these two notions of finer/coarser?