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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: enter image description here

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 $\mathbb{R}^n.$

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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 $\Bbb{R}^2$.

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  • $\begingroup$ 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. $\endgroup$
    – theQman
    Dec 7, 2020 at 1:31
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    $\begingroup$ What is your definition of manifold $\endgroup$ Dec 7, 2020 at 1:50
  • $\begingroup$ @theQman I second Alekos's sentiment - you kind of need charts in order to define what a manifold is. What is the definition of manifold you are using? $\endgroup$
    – K.defaoite
    Dec 7, 2020 at 3:49
  • $\begingroup$ I have added the definition to the question. $\endgroup$
    – theQman
    Dec 7, 2020 at 14:18
  • $\begingroup$ 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 $U_2$. $\endgroup$ Dec 7, 2020 at 16:17

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