Let be a union of a countable number of "lines" in .
Prove that is connected.
This seems intuitively true - take two points . Look at the line . 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, moreso considering that it's a question from a previous exam. Any ideas/hints are welcome.