Product space between circle and disk

Question

What is the product space between the unit circle in the complex plane and the unit disk in the complex plane?

Isn't it a torus? That would be my intuitive answer if you would say the product space between the unit circle and the unit disk in $\mathbb{R}^2$, but there is no three dimensional complex space is there?

Thanks

Answer 1

The product of two circles is a torus: $T^2 = S^1 \times S^1$ (embedded in 4-dimensional space).

The product of a circle and a disk is a solid torus (embedded in 4-dimensional space). The torus $T^2$ is this solid torus' surface.