 Solution

Define $F:[0,1]\to \mathbb{C}$ by $F(s)=s{e}^{2\pi i/s}$ for $s>0$ and $F(0)=0$. It is continuous away from zero, and $|F(s)-F(0)|=s\to 0$, so it is continuous at zero too. Its image is exactly $X\cup \{0\}$, using $s=1/t$. The interval is compact, so its continuous image is compact.

For any two image points $F(s_0),F(s_1)$, the explicit pathis continuous and joins them. Thus the reciprocal spiral with its endpoint is also path-connected. Infinite winding causes no discontinuity at the endpoint because its radius tends to zero.