How can I visualize the Cartesian product of sets? I saw the question asking about intervals and simple Cartesian products, but how can I visualize things like $S^1 \times S^1$, the Cartesian product of the unit 1-sphere? I understand that this is a Torus, but what should my thought process be here?
 A: A pair of angles $\theta$ and $\phi$ determine an element $(e^{i\theta},e^{i\phi})$ in $S^1\times S^1$, which in order they can be consider that give a position in the torus.
This is achieved mapping via $(e^{i\theta},e^{i\phi})\longrightarrow\left((2+\cos\theta)\cos\phi,(2+\cos\theta)\sin\phi,\sin\theta\right)$
In the picture, the blue point on the torus is specified by the two angles in green.

A: You can think of $X \times Y$ as substituting a copy of $Y$ for each $x \in X$. For spaces (and other kinds of structures, e.g. orderings), each copy $Y_x = Y$ of $Y$ retains its notion of "nearness", but the notion of nearness on $X$ is also retained: for $x, ', x'' \in X$, if $x$ is nearer to $x'$ than to $x''$, and if $y \in Y$, then the copy $(x,y)$ in $Y_x$ will be nearer to the copy $(x', y)$ in $Y_{x'}$ than to the copy $(x'',y)$ in $Y_{x''}$.
In the case of $S^1 \times S^1$, you're replacing every point on the (first) circle with a copy of the second space (another unit circle), taking care to define the neighborhoods so that "nearness" is as described before.
Similarly, orderings $X, Y$ give rise to an ordering on $X \times Y$ – the dictionary or lexicographic order. Again you can think of it as substituting a copy of $Y$ for each $x \in X$. For example, where $3$ is the 3-element set $\{0,1,2\}$ with the usual ordering, then $3 \times \mathbb{N}$ with the dictionary order looks like this:
$$
0, 1, 2, \dots, 0', 1', 2', \dots, 0'', 1'', 2'', \dots \text{.}
$$
