A set that it is uncountable, has measure zero, and is not compact

Question

I want a example of a set that it is uncountable and has measure zero and not compact? Cantor set has these properties except not compactness.

Answer 1

Recall the construction of the Cantor set. At each step we remove open intervals, now remove closed intervals.

The intersection is still non-empty, but the result is something homeomorphic to the irrational numbers, or more generally, Baire space.

This is not a compact set, since we can show that the points removed are in the closure of this new set, and thus it is not closed. However as a subset of the Cantor set it still has measure zero.

Answer 2

Just delete a point, say $0$, from the Cantor set and you'll get a set with the desired properties. In fact, since the Cantor set is perfect, any point will do.

Answer 3

Let $B$ be the union of translates of the Cantor set by every integer $n$. Then $B$ is uncountable, has measure $0$. It is unbounded so not compact.