# How do we know that $S^1 \times S^1$ is a torus? [closed]

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?

• To me, $S^1\times S^1$ IS the definition of a torus. What is yours?
– RKD
Mar 21 '16 at 18:29
• This essentially follows from this picture: de.wikipedia.org/wiki/Datei:TorusAsSquare.svg If you start with a square and identify the $A$'s, you obtain a cylinder. If you identify the $B$'s, you obtain a torus. Further, it is easy to see that the result of such identification is $S^1\times S^1$, as both $A$ and $B$ are now circles. Mar 21 '16 at 18:45
• I don't understand why this question was put on hold. It was very clear to me what OP was asking. Maybe it's because I had the exact same thought when I first learned about the torus from a topological perspective. "$S^1 \times S^1$ is the definition of a torus" doesn't work for new learners. The OP wants to know how that definition makes sense. In other words, how is it that the torus can be described entirely by points $(\alpha, \beta) \in S^1 \times S^1$? What do the coordinates of those points represent?
– user307169
Mar 22 '16 at 12:50
• @tilper I thought there was no such thing as a dumb question. Unfortunately, this site has come down to questions being closed, or put off topic, because they are "obvious" to some. You can't just ask a simple question that you are curious about anymore. shame. Jun 16 '18 at 20:16
• @AlJebr, well, I've heard some dumb questions for sure (which usually result from people just not paying attention, not from a lack of understanding), but this isn't one of them. Agreed on your second point and that's why I don't come around here much anymore.
– user307169
Jun 21 '18 at 15:02

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 $\beta \in [0,2\pi) = S^1$.