What is a refinement useful for? From Wikipedia: A refinement of a cover $C$ of a topological space $X$ is a new cover $D$ of $X$ such that every set in $D$ is contained in some set in $C$. Formally,

I find this definition easy to understand, but I am interested in why we need to introduce such term.
One way to use it is for defining the (Lebesgue) covering dimension - the smallest number $n$ such that for every cover, there is a refinement in which every point in $X$ lies in the intersection of no more than $n + 1$ covering sets.
The only intuition I have is that the covering dimension tells how well can we cover the space by "non-overlapping" sets. But I still do not see, why we need to introduce the refinement.
Could you help me gain some intuition about refinement, and possibly also what role does it plays in the definition of the covering dimension?
Thank you.

If we imagine a general open covering of (say) the real line, it could be very messy: the open sets could be disconnected, many of the open sets could be identical or nearly identical, and so on. Thinking about refinements allows us to clean up the covering so that it takes the "nicest" form: a sequence of open intervals overlapping slightly at the ends. For example, when an open set is disconnected, we can break it into its interval components; when two of the open sets are identical, we can just take one of them.
That "nicest" form then suggests a canonical structure to the space: in this case, a refinement in which at most two open sets overlap at any point.
