= Solution
For the <torus> quotient, a point has the same image as a point of $B(u,r)$ precisely when they differ by an integer lattice vector. Therefore
$$
\boxed{\pi^{-1}(U)=\bigcup_{m\in\mathbb Z^2}B(u+m,r).}
$$
This union is open in $\mathbb R^2$, so $U$ is open by the <quotient topology>. Distinct integer lattice vectors are at least distance one apart, whereas $2r<1$. Thus these open balls are disjoint, with a positive gap between any two.
Each ball is connected, since it is convex. It is both open and closed relative to the displayed union: its complement there is the union of all the other open balls. No <connected component> can meet two such relatively separated pieces. Consequently \b[the components are exactly the infinitely many balls $B(u+m,r)$]. Translation by $-m$ is a <homeomorphism> from each component onto $B(u,r)$. The statement uses the intended positive radius $0<r<1/2$.
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