= Spiral accumulating on a circle
{title2=$Y=\{(1+t^{-1})e^{2\pi it}:t\ge1\}$}
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>.
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