Let $Circ$ be the topological group

$(\{z\in \mathbb{C} : \overline{z}\cdot z = 1\},\cdot , \{U\in 2^{\{z\in \mathbb{C} : \; \overline{z}\cdot z \, = \, 1\}} : \{z\in \mathbb{C} : \overline{z}\cdot z = 1\}-U $ is closed in $ \mathbb{C}\})$.

Let $(G,\star,T)$ be a topological group such that $(G,T)$ is homeomorphic to $(\{z\in \mathbb{C} : \overline{z}\cdot z = 1\}, \{U\in 2^{\{z\in \mathbb{C} : \; \overline{z}\cdot z \, = \, 1\}} : \{z\in \mathbb{C} : \overline{z}\cdot z = 1\}-U $ is closed in $ \mathbb{C}\})$.

Does it follow that

- $(G,\star)$ is isomorphic to $(\{z\in \mathbb{C} : \overline{z}\cdot z = 1\},\cdot)$?
- $(G,\star,T)$ is homeomorphically isomorphic to $Circ$?

threegroup structures, as pointed out to me by Greg Kuperberg. – Allen Knutson Oct 28 '10 at 18:28