# Proof that $S^1$ is a manifold using projections

I'm trying to understand what seems like it should be a simple proof in Spivak's text on differential geometry. In order to show that the circle, $$S^1$$, is indeed a manifold, he projects $$S^1 \setminus \{(0,1)\}$$ onto $$\Bbb{R} \times \{-1\}$$, as described in this image:

I can intuitively see how this gives a homeomorphism between $$S^1 \setminus \{(0,1)\}$$ and $$\Bbb{R} \times \{-1\}$$.

Next, he says that the point $$(0,1)$$ can be handled similarly by projecting it on $$\Bbb{R} \times \{1\}$$. How does this projection work? Is the point $$(0,1)$$ just sent to itself?

My bigger question is, how are these two arguments combined to show that $$S^1$$ is indeed a 1-manifold? The first part shows "most" of $$S^1$$ being homeomorphic to $$\Bbb{R}^1,$$ and the second part shows the "rest" of $$S^1$$ being homeomorphic to... something? Then what?

For reference, here's the definition of manifold in the text: A manifold is a metric space $$M$$ with the following property: If $$x \in M,$$ then there is some neighborhood $$U$$ of $$x$$ and some integer $$n \ge 0$$ such that $$U$$ is homeomorphic to $$\mathbb{R}^n.$$

This is sometimes called stereographic projection "from the north pole" or something like that, thinking of $$(0,1)$$ as the north pole of the circle. We can analogously defined stereographic projection "from the south pole" by the same type of construction, except the rays emanate from $$(0,-1)$$ and land on the line $$\Bbb{R}\times \{1\}\subseteq \Bbb{R}^2.$$ Using this projection map, $$(0,1)$$ is indeed sent to itself, and we get a diffeomorphism $$S^1\setminus \{(0,-1)\}\to \Bbb{R}$$. Combine this with the other diffeomorphism $$S^1\setminus \{(0,1)\}\to \Bbb{R}$$ and we get an atlas of charts for $$S^1$$.
What we have then shown is that there exists an open cover $$\{U_1,U_2\}$$ of $$S^1$$ with diffeomorphisms $$\phi_i:U_i\to \Bbb{R}$$. This proves that $$S^1\subseteq \Bbb{R}^2$$ is indeed a submanifold of $$\Bbb{R}^2$$.
• I'm not familiar with atlas of charts yet... The idea was to more or less see that $S^1$ is a manifold using pretty much only the definition of manifold. Dec 7, 2020 at 1:31
• An atlas is the open cover of sets $U$ that you have in your definition. I.e. an open cover of $M$ by opens $U$ homeomorphic (in this case diffeomorphic) to $\Bbb{R}^n$ for some $n$. Our $U$'s here are $U_1$ and $U_2$. Dec 7, 2020 at 16:17