On pg. 48 of Hatcher's Algebraic Topology, the author writes that the $3$-sphere $S^3$ can be thought of as the union of two solid torus.
First a formal reasoning is given which is
$S^3=\partial D^4 = \partial(D^2\times D^2) = (\partial D^2\times D^2)\cup (D^2\times \partial D^2)$
This is clear. But then the author gives a geometric interpretation which is as follows.
Think of $S^3$ as the one point compactification of $\mathbf R^3$. Let $T=S^1\times D^2$ be the first solid torus. The second solid torus is the closure of the complement of $T$ in $\mathbf R^3$ along with the compatification point at infinity.
The statement in italics is not clear to me. How do we see this as a solid torus?