Spiral accumulating on a circle ($Y=\{(1+t^{-1})e^{2\pi it}:t\ge 1\}$)

The closure of the displayed connected spiral adds the unit circle. Its union with the closed unit disk is therefore connected but not path-connected. A path from the spiral to the disk would have a first contact time. Before contact its reciprocal radial offset defines a continuous parameter tending to infinity. The intermediate value theorem then forces sequences of path points with parameters integer and half-integer, approaching opposite circle points. This contradicts continuity at the first contact. Thus the spiral and disk form separate path-connected components.