What does S^1 X S^1 really mean?

[pivoxa15]

Torus = S^1 X S^1 but what if someone was only shown S^1 X S^1, what would it mean? Would they intepret it as a torous? I know I wouldn't.

So what is S^1 X S^1 defined as? If you say a torous then what is a torus? Don't just draw a picture.

 

[HallsofIvy]

S1 is a circle. S1 X S1 is a Cartesian product. I would interpret it as the set of all ordered pairs, each pair being two points on a circle. If I were then to interpret the two points, (p1, p2), as coordinates for a point on a surface- go around a circle to the point given by p1, then at right angles around a circle to the point given by p2- yes, that would look like a torus to me.

What is a torus without drawing a picture? A torus is a doughnut, of course- a glazed doughnut just out of the frier! Yum! (One good bite and it is homeomorphic to a ball.)

 

[pivoxa15]

I think I see. RXR is the plane. S^1XS^1 does map out a torus which clearly is a surface with an empty inside. Take a bite and it is homeomorphic to a cylinder, not a ball!?