# Is a set of measure zero in $\mathbb{R}$ totally disconnected?

Let $$M \subset \mathbb{R}$$ be a nonempty set of Lebesgue measure zero. Does it follow that $$M$$ is totally disconnected in the sense that for any $$x, with $$x,y\in M,$$ there exists $$z\notin M$$ such that $$x?

I think the answer to the questions is yes, since otherwise one could argue by contradiction and say that then there is an interval contained in $$M$$ and hence it cannot have measure zero.

Is the reasoning above sound? Also just as side question, connectedness of nonempty sets does not imply positive measure in $$\mathbb{R}^n$$, since a line in $$\mathbb{R}^2$$ is connected and has measure zero, right?

Thank you for your time and appreciate any feedback.

• Hint: Can a set of measure zero contain, as a subset, an interval that contains more than one point? Follow-up Hint: The measure of an interval containing more than one point is {negative, zero, positive} (pick one), and measure is a monotone set-function. – Dave L. Renfro Feb 16 at 9:11
• @DaveL.Renfro I think it cannot, since that would contradict the set having measure zero. Please correct me if I am wrong. – Gaby Alfonso Feb 16 at 9:13
• You are correct. This is one of those questions that I imagine initially seems more difficult than it is, possibly because all the Cantor set weirdness makes you extra cautious and the notion "totally disconnected" is possibly something you deal with a lot more in topology than in an analysis course. – Dave L. Renfro Feb 16 at 9:16

You are right: since, in $$\mathbb R$$, the non-empty connected sets are the intervals and since the non-degenerated intervals have positive measure, a set with measure $$0$$ cannot contain a non-degenerated interval and therefore it is totally disconnected.
You are are right too about $$\mathbb{R}^n$$, when $$n>1$$.