This seems rather simple, but just curious about the following definition (pulled from Lee, but definitely standard):
Given an open cover $\mathcal{U}$ of $X$, another open cover $\mathcal{V}$ is called a refinement of $\mathcal{U}$ if for each $V\in \mathcal{V}$ there exists some $U\in \mathcal{U}$ such that $V\subset U$.
Why not the following definition:
Given an open cover $\mathcal{U}$ of $X$, another open cover $\mathcal{V}$ is called a refinement of $\mathcal{U}$ if for each $U\in \mathcal{U}$ there exists some $V\in \mathcal{V}$ such that $V\subset U$.
To me this second 'definition' seems more natural, is this important? I guess these defintions are not equivalent but how bad is this definition of mine?
Your definition, while initially tempting, would make $\{X,\emptyset\}$ a refinement of every open cover! Basically, we need to be sure that every piece of a refinement is small relative to the original cover, and that's what the usual definition gets that yours doesn't. On the other hand, the usual definition does permit, say, $\{X\}$ as a refinement of the cover by all opens: roughly, a refinement doesn't have to be "fine" relative to the small parts of the original cover, but just relative to the big parts. It's a general principle that the small sets in an open cover (i.e. those contained in another element of the cover) are redundant and unimportant.