Circle with a spiral

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Let $S^1 \subset \mathbb{R}^2$ denote the unit circle parametrized in polar coordinates as $S^1=\{ (r,\theta) \colon r=1;\theta \geq 0 \}$. Let $S \subset \mathbb{R}^2$ denote the set (defined in polar coordinates) $S=\{(r,\theta) \colon r = 1+\dfrac{1}{1+\theta}; \theta \geq 0 \}$. The circle with a spiral is the set $C=S^1 \cup S$.

Circlewithspiralfortheta=0totheta=40.png

Properties

Cone over circle with spiral homeomorphic to hyperspace of continua of circle with spiral