In lectures we defined:
Let $X$ be a topological space, $Y \subset X$ a subset. A collection $\mathcal{A} \subset \mathcal{P}(X)$ is a cover of $Y$ by sets open in X if each element of $\mathcal{A}$ is an open set in $X$ and $Y \subset \cup_{A \in \mathcal{A}}A$.
We then went on to a proposition:
Let $Y$ be a subset of $X$. Then $Y$ is compact if and only if every cover of $Y$ by subsets open in $X$ has a finite subcover.
I think the "subsets" part in the proposition should actually be "sets", but I want to make sure since we could end up with something different otherwise.
(This particular lecturer takes a while to answer emails so I'm asking here instead.)
Saying that $B$ is a subset of $X$ does not mean that $B$ is not equal to $X$. (We call those proper subsets.) Hence, it does not make any difference whether you call them sets or subsets.