Circle Times Circle

The product of the real line with itself can be visualized as assigning each point a copy of the real line. This gives the real plane.

The product of the real line with the real plane can be visualized as assigning each point a copy of the real plane. This gives the real space.

The product of a circle with itself can be visualized as assigning each point on the circle a copy of the circle. This gives a torus.