I know a torus as the product of circles/ovals. So a torus in $\mathbb{R}^3$ can be $S^1 \times S^1$ or the product of cirles with different radii. But is there a concrete definition of a torus? I didn't found any!
definition of a (topological) torus
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general-topology
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0How is $S^1\times S^1$ not concrete? – 2017-01-23
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0$S^1 \times S^1$ is just one special torus. With concrete definition I mean something like: A square is a plane with 4 corners and 4 90-degree-angels where all edges have the same length – 2017-01-23
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0A square is a metric object, because it's definition depends on lengths and angles. A torus is a topological object and there's only one torus for each dimension, namely $(S^1)^n$. If you want to embed a ($2$-dimensional) torus in $\Bbb R^3$ you know what you get: a doghnut's skin. – 2017-01-23
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0I thought you can vary the radii of the circles and then you have a torus too. Or do you mean, they are topologically all the same and so there is just one torus? – 2017-01-23
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0I mean just that, yes. – 2017-01-23