Question

How would you correspond in a bijective way all the points of a circle with holes in it with a straight line?

More specifically, explain the process that you would follow to build the bijective function that makes the points of the set {(x,y): x^2+y^2=1}-{(0,1)} correspond to the real line.

Answer 1

To create a bijective function between a punctured circle and the real line, a stereographic projection is used. By connecting points on the unit circle to the 'north pole', the resulting intersections with the x-axis provide a one-to-one correspondence with the real line.

Step-by-step explanation:

To establish a bijective function that corresponds all points of the unit circle with a hole at (0,1) to the real line, one can use a stereographic projection. The process involves projecting every point on the circle (minus the hole) from the north pole of the circle (the point with the hole) onto an extended real line.

Here's a step-by-step explanation of setting up the bijection:

1. Imagine the unit circle in the coordinate plane, and above the circle, place a point that corresponds to the north pole (0,2) in this context, as the hole is at (0,1).
2. For each point P on the circle (except the north pole), draw a line connecting P to the north pole. Extend this line until it intersects the x-axis, which we will use as our real line. This point of intersection is the corresponding point on the real line to P on the circle.
3. Formally, for a point (x,y) on the circle, the stereographic projection maps it to the point X on the real line, where X is obtained by the intersection mentioned above. The equation for this projection can be derived geometrically.
4. Conversely, every point on the real line corresponds to exactly one point on the circle, making the function bijective.

Essentially, we are transforming the circular coordinates into Cartesian coordinates via this projection, ensuring a one-to-one and onto mapping between the two sets.