= Solution
Define $F:[0,1]\to\mathbb C$ by $F(s)=se^{2\pi i/s}$ for $s>0$ and $F(0)=0$. It is continuous away from zero, and $|F(s)-F(0)|=s\to0$, 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 \b[compact].
For any two image points $F(s_0),F(s_1)$, the explicit path
$$
\alpha(u)=F((1-u)s_0+us_1),\qquad0\le u\le1,
$$
is continuous and joins them. Thus the <reciprocal spiral with its endpoint> is also \b[<path-connected>]. Infinite winding causes no discontinuity at the endpoint because its radius tends to zero.
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