I am trying to understand the above proposition. So according to the definition of $\text{dim}\, X$, $\mathcal{W}$ is a refinement of the open cover. What I don't understand is why is it that any point $x$ in $X$ must be in some member of $\mathcal{W}$? Does this imply that the refinement $\mathcal{W}$ is also a cover of $X$? Why is this not mentioned in the definition?
As Codenotti already pointed out, the relation of refinement is defined between covers of a space. So it's sort of implicit in the use of the word refinement, as it were.