Why is $x^2+y^2=r^2$ a $1$-sphere and not a $2$-sphere?

Question

According to Wikipedia a 1-sphere is a circle and a 2-sphere is the ordinary 3-dimensional sphere.

But why are not:

• $x$ a $0$-sphere (just one point)?

• $(x,y)$ a $1$-sphere (a pair of points)?

• $x^2+y^2=r^2$ a $2$-sphere (due to two variables)?

• $x^2+y^2+z^2=r^2$ a $3$-sphere (due to three variables)?

and so on ...

Thanks in advance!

Answer 1

1 is the dimension of the circle itself. You can identify a point on the circle by specifying one number, say the angle. (Formally speaking, the circle is a 1-dimensional manifold.) You can imagine the circle as something you get by distorting a line or a piece of string, which are both "manifestly" 1-dimesional.

2 is the dimension of the space it sits in.