 Solution

Each vertical line segment over a rational coordinate meets the horizontal line at height one. A point on such a segment can move vertically to the top, then horizontally along that line; points already at height one use only the horizontal motion. Concatenating these motions joins any two points.

Thus the rational comb with a deleted attachment point construction, before deletion, is path-connected and hence connected. The infinitely many vertical pieces cause no difficulty because each individual path uses only finitely many segments.