Isomorphism between $\mathbb{T}$ and $\mathbb{R} \oplus \mathbb{Q}/\mathbb{Z}$.

I'm not sure how to make it work via cardinal arithmetic, but here is a direct proof.

First, a vector spaces over $\mathbb{Q}$, both $\mathbb{R}$ and $\mathbb{R}\oplus\mathbb{Q}$ have dimension $|\mathbb{R}|$, so they are isomorphic.

Choose an isomoprhism $f:\mathbb{R}\rightarrow \mathbb{R}\oplus\mathbb{Q}$ once and for all. Let $z= f^{-1}(0,1)$. Since $f$ is a homormorphism, $z\neq 0$, so $\langle z\rangle$ is isomorphic to $\mathbb{Z}$.

Then $f$ induces an isomorphism between $\mathbb{R}/\langle z\rangle \cong \mathbb{R}/\mathbb{Z}\cong S^1$ and $\mathbb{R}\oplus \mathbb{Q}/\langle (0,1)\rangle = \mathbb{R}\oplus (\mathbb{Q}/\mathbb{Z})$.