How can I imagine $S^1 \times D^2$

Question

$S^1 \times S^1$ can be imagined as a torus in $\mathbb{R}^3$.

So is $S^1 \times D^2$ a filled torus? Because $D^2$ is a filled circle?

Is this right?

Answer 1

Every time you have a, say bounded, $A\subset{\Bbb R}^n$ you can "picture" $A\times S^1$ simply adding an extra $(n+1)$-st coordinate and rotating $A$ around the extra axis (you need to be careful about shifting $A$ far enough the extra axis to avoid self-intersections).

In your case things get simple enough because $D\subset{\Bbb R}^2$ and it's easy to picture the result.