= 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 \b[<path-connected> and hence <connected>]. The infinitely many vertical pieces cause no difficulty because each individual <path> uses only finitely many segments.
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