# S1 isomorphic to S1/Z2?

1. Nov 11, 2009

### variety

S1 isomorphic to S1/Z2??

1. The problem statement, all variables and given/known data
This isn't really a homework problem, but whatever. So in my textbook, it says that "of course $$S^1/Z_2$$ is isomorphic to $$S^1$$." I want to know why.

2. Relevant equations

3. The attempt at a solution
Is $$S^1$$ even a group? Well I guess it can be a group, with multiplication as the operation ($$S^1$$ viewed as a subset of $$C$$). But then the only map I can think of between the two is the projection $$p:S^1\rightarrow S^1/Z_2$$, where $$x\in S^1$$ is mapped to its obit under the action of $$Z_2$$. But this map is clearly not injective since $$p(x)=p(-x)$$ for any $$x \in S^1$$.

If this is a typo in my book, then are the sets homeomorphic (maybe they wanted to use the symbol for homeomorphism instead of isomorphism)? What then is the map between them?