 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.