Is the Riemann sphere really just the entire complex plane where you add a "single point" at infinity? It's well known that the Riemann Sphere is defined as the complex plane plus a point at infinity.
Question: Is the point at infinity really a single point, or is it a circle that has dimension 0?
If we make the replacement $z \to \frac{1}{z}$ then we can say that the origin represents the point at infinity.  But this is really not accurate, since the origin under this mapping got deleted, and it's necessary to treat every possible angle of approach as a distinct path.  This corresponds to the fact that starting from any finite part of the complex plane, there are an infinite number of ways to move in a straight line off to infinity. [1]
The problem of treating infinity as a single point becomes apparent when we take some polynomial which is well defined everywhere in the entire complex plane, and only has a single singularity at infinity, and make the transformation
$$z \to \frac{1}{z}$$
Then what happens is our polynomial just got turned into a rational function that has poles which are all over the complex plane, and whereas the polynomial can approach infinity from any direction and it's only singularity is at the point at infinity, when we take $z \to \frac{1}{z}$ the singularity at infinity disappears and is replaced by poles which are not all at the same point.
This makes perfect sense (to me) when the point at infinity is a circle with dimension 0.  Although the concept of "a circle with dimension 0" is something I just made up, and which is not well defined, my intuition tells me that there is a topological distinction between a single point and a circle with dimension 0.  Is there any such concept as a circle with dimension 0, or is that a bad way of thinking about this?
I want a circle with dimension 0 to be the point of tangency between two spheres that only touch at a single point.  In other words, its not just a single point because it has an extra dimension to play around with, but the intersection just looks like a single point, if that makes sense.
This issue about the difference between the Riemann sphere and the standard projective geometry approach has been bothering me for over 10 years, and any help clarifying my confusion here is very much appreciated.
Thanks!
[1] Modern Anaylysis, Whittaker & Watson, 4th edition, 5.62 The "point at infinity"
 A: I have never really thought of the Riemann Sphere as anything more than a abstract tool. It helps us better understand why particular complex functions behave the way they do. Take for example, LFTs (linear fractional transformations). Now it is a well known theorem that LFTs map "circles and lines" to "circles and lines". However this feels cumbersome and there is a certain beauty lost in trying to distinguish lines from circles, or in attempting to figure which lines go to which circles or which lines and visa-versa. However, a particular clarity is found when LFTs are placed on the Riemann Sphere; and we suddenly realize that the statement was in error, really LFTs map circles to circles, that's it. There are no lines on the Riemann Sphere; lines in the complex plane are simply the special case of circles on the Riemann Sphere that happen to go through $\infty$. Now doesn't that sound nice to say, "LFTs map circles to circles".
This kind of abstract extension of a space "to complete it" is done all the time. A similar example would be Elliptic Curve Geometry. Without going into the nuances, assume for a moment that their exist curves (symmetric about the x-axis) with the property that given any two points on the curve and the tangent line between them, then there is guaranteed to exist a third point on the curve that passes through the tangent line. We can then define an algebraic "addition" on the curve, by $A+B=C$ if $C$ is the reflective point across the x-axis of the third point that intersects the tangent line between $A$ and $B$. This works everywhere, except one small caveat. What happens if $A$ and $B$ are exactly vertical from each other, then algebraically they are apparently $A$ and $-A$, opposites of each other. But there is no third point that passes through a vertical line on an Elliptic Curve. So how do we resolve that in our mind? The geometry shows there is no third point, yet logically, $A+(-A)$ should be $0$. The solution is that $0$ is really the point at $\infty$, and so on an Elliptic Curve, the abstraction that $A+\infty=\infty+A = A$ for any $A$, makes sense geometrically; thus $\infty$ is the zero of the abstract group.
