Originally Posted by

**Defunkt** Let $\displaystyle A$ be a union of a countable number of "lines" in $\displaystyle \mathbb{R}^3$.

Prove that $\displaystyle B := \mathbb{R}^3 - A$ is connected.

This seems intuitively true - take two points $\displaystyle a, b \in B$. Look at the line $\displaystyle [a, b] = \{ ta + (1-t)b \ : \ t \in [0,1] \}$. If a point on the line intersects one of the lines in A, then simply deviate the connecting path a little bit, such that you don't intersect with the next closest line.

Making this rigorous, however, seems a bit painful, more so considering that it's a question from a previous exam. Any ideas/hints are welcome.