Rational comb with a connecting spine
= Rational comb with a connecting spine
The union of $\{0\}\times[0,1]$ with horizontal segments $[0,1]\times\{q\}$ for rational $q\in[0,1]$ is <path-connected>. Each point lies either on the vertical spine or on a segment meeting it. A path travels horizontally to the spine, vertically along it, and horizontally to its destination. This contrasts with disconnected rational-slope rays whose common origin has been excluded.