Math 104 Assignment 4 common mistakes

• Some confusion about subcovers: people tried to construct finite subcovers (or double sub-covers) but were adding open sets which were not part of the original cover. The "sub" part of "sub-cover" is important.
• Some people assumed that a double cover must contain two disjoint covers, but this is not true. Consider the set $\set{1,2,3}$, and the open double cover $\set{\set{1,2}, \set{2,3}, \set{3,1}}$. (Of course, this double cover is already finite, but one can manipulate this idea to find a compact set and a double cover which has no finite double sub-cover.)
• Some people tried to apply the Heine-Borel theorem to sets which were not subsets of $\R^d$, which is incorrect. In at least some of these cases, they merely wanted the statement that compact sets are closed and bounded in any metric space, which is true, but is not the Heine-Borel theorem.