Why Can't Two Functions Cover the Unit Circle?

[Math Amateur]

I am reading the book: "Vector Calculus, Linear Algebra and Differential Forms" (Fourth Edition) by John H Hubbard and Barbara Burke Hubbard.

I am currently focused on Section 3.1: Manifolds ...

I need some help in order to understand Example 3.1.3 ... ...

Example 3.1.3 reads as follows:View attachment 8633In the above text from H&H we read the following:

"Here we need the graphs of four functions to cover the entire circle ... "My question is as follows:

Why can we not cover the unit circle with the following two functions:

\(\displaystyle y = \sqrt{ 1 - x^2 }\) where \(\displaystyle -1 \le x \le 1\)

and

\(\displaystyle y = - \sqrt{ 1 - x^2 }\) where \(\displaystyle -1 \lt x \lt 1\)

I must be misunderstanding something ...

Hope someone can help ...

Peter

 

[Opalg]

In the context of the Hubbards' book, one wants to cover a manifold (the circle, in this example) with open patches. So each point of the manifold must lie in the interior of one of the patches. The easiest way to do that in this case is to use four patches.

 

[Math Amateur]

In the context of the Hubbards' book, one wants to cover a manifold (the circle, in this example) with open patches. So each point of the manifold must lie in the interior of one of the patches. The easiest way to do that in this case is to use four patches.

Thanks Opalg ...

That makes the issue clear ... grateful for that!

Peter