Let $F : I \times I \rightarrow X$ be a continuous map and U, V be two open covers of X. Then Lesbegue lemma says that there exist partitions of I which are $0=s(0)<...<1=s(m)$ and $0=t(0)<...<1=t(n)$ such that each rectangle $R(ij) = [s(i), s(i+1)] \times [t(j), t(j+1)]$ of $I^2$ is mapped into $U$ or $V$.
Now the question is : Is it possible to claim that the edges of R($ij$) are included in the intersection of $U$ and $V$? And if possible, how to do it?
I know the process when $F$ is just a path from $I$ to $X$, but can't extend the method when $F$ is given as above. Could anyone help me?