How do we know that $S^1 \times S^1$ is a torus?

Question

I was reading a Dugunjis "Topology" and in the first chapter it is said that $S^1 \times S^1$ (Cartesian product) where $S^1$ is a unit sphere in 2 dimension is a torus.

How can this be explained?

Answer 1

This would work a lot better if we could have this discussion in person with visual aids, but here's my attempt.

Think of $S^1$ as $[0,2\pi)$, where the elements represent angles in standard form. Then any point on the torus can be uniquely described by a point $(\alpha, \beta) \in S^1 \times S^1$.

Imagine you have a typical donut resting on a table. Take a very small "disk" out of it, like the disk of radius $a$ in the picture here:

You can uniquely identify any point on the boundary of that disk with some angle $\alpha \in [0,2\pi) = S^1.$

So what's the $\beta$ represent? Going back to our donut sitting on the table, imagine now the table is the $xy$-plane and the origin is at the center of the donut hole. Remember that "disk" we sliced out in the previous paragraph? Well, if we view our donut straight from above, then that "disk" will just look like a straight line segment. Imagine extending this line segment so it reaches the origin. This line segment now makes an angle with the positive $x$-axis. Call this angle $\beta$, and note that we can always take $\beta \in [0,2\pi) = S^1$.