How do we reconcile, probably non set-theoretically, that $S^1\times S^1$ is a torus, and $S^2$ is a sphere?

Question

The sphere is $S^2$ and the Torus is $S^1\times S^1$. I thought set theoretically atleast, that $X\times X = \{(x,y): x\in X, y\in X\} = X^2$. Is this notion that $T=S^1\times S^1$ and the sphere is $S^2$ where these aren't the same, a non-set theoretical notion?

Answer 1

It's just a different notation. Most of the time, $X^2$ means $X\times X$, but in the special case $S^2$, it means something different. More generally, when talking about geometric objects, it is common for a superscript to refer to the "dimension" of the object, rather than being an exponent. It can be confusing sometimes, but usually in context it is easy to understand which meaning is intended by a superscript.