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 $x<y$, with $x,y\in M,$ there exists $z\notin M$ such that $x<z<y$?
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.
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 $n>1$.