# Closed rectangle contained in set that does not have measure zero.

I was doing some analysis when it occurred to me that if I knew the answer to the following question many of the exercises would be simpler. However, I haven't been able to make much headway. Any help will be appreciated.

Question: If I have a set $S \subset \mathbb{R}^n$ such that $S$ is not of measure 0, is it always true that there exists some closed rectangle $C$ such that $C \subseteq S$?

• The answer is no, $\mathbb{R}-\mathbb{Q}$ being an example (as indicated in the answers). But there are some important instances where the answer is true that are helpful to keep in mind. For instance, the set on which a continuous function is positive (or negative) has positive measure, being that it contains a $k$-cell (rectangle). – Sargera Jun 19 '13 at 23:31

No. Think of $\mathbb R \backslash \mathbb Q$.
Counterexample: $[0,1]-\mathbb{Q}$ has a non zero measure, yet it does not contain an interval.
• Doesn't it contain $]1/3, 2/3[$? – Patrick Da Silva Jun 19 '13 at 23:20