# Euclidean space as a sum of non-intersecting and non-parrarel lines

Is it possible to represent $\mathbb{R^{3}}$ as a union of countable non-intersecting and non-coplanar lines?

• – hot_queen Jan 23 '14 at 16:23

No, they have measure 0, thus so does their countable union. Whereas $\mathbb R^3$ has infinite measure.
Also by the baire category theorem a complete metric space cannot be written as the countable union of closed, nowhere-dense sets and lines are nowhere dense in $\mathbb R^3$.