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

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In summary, S^1 X S^1 is a Cartesian product that can be interpreted as the set of all ordered pairs of points on a circle. When these points are used as coordinates on a surface, it resembles a torus. A torus is a doughnut shape, which is homeomorphic to a ball when a bite is taken. However, when using S^1 X S^1 to map out a torus, it actually appears as a surface with an empty inside and is homeomorphic to a cylinder when a bite is taken.
  • #1
pivoxa15
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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.
 
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  • #2
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.)
 
  • #3
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!?
 

1. What is S^1 X S^1?

S^1 X S^1 is the Cartesian product of two circles, also known as the torus. It can be visualized as a doughnut shape with a hole in the middle.

2. How is S^1 X S^1 different from S^1?

S^1 is a single circle, while S^1 X S^1 is the product of two circles. This means that S^1 X S^1 has two dimensions, while S^1 only has one.

3. What does the ^ symbol mean in S^1 X S^1?

The ^ symbol indicates exponentiation, meaning that S^1 X S^1 is the result of raising S^1 to the power of 2. In this case, it represents the two dimensions of the torus.

4. How is S^1 X S^1 related to topology?

S^1 X S^1 is a topological space, meaning it is a set of points with a defined structure. It is commonly used in topology to explore concepts such as compactness, connectedness, and orientability.

5. What are some real-world examples of S^1 X S^1?

S^1 X S^1 can be found in many everyday objects, such as a rubber band twisted into a figure-eight shape or a coffee mug with a handle. It can also be seen in nature, such as a donut-shaped galaxy or the shape of a bicycle tire.

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