= Solution
Under
$$
\phi:\mathbb A_K^\times\longrightarrow\mathbb A_K\times\mathbb A_K,
\qquad
x\longmapsto(x,x^{-1}),
$$
the preimage of a basic product neighbourhood $U\times V$ is
$$
U\cap V^{-1}.
$$
At almost all finite places, $U_v=V_v=\mathcal O_v$, and the condition $x_v,x_v^{-1}\in\mathcal O_v$ is exactly $x_v\in\mathcal O_v^\times$. Thus these preimages form the usual restricted-product basis for $\mathbb I_K$. Conversely, every basic idele neighbourhood is obtained by choosing suitable $U_v$ and $V_v$ locally. Hence the <restricted product topology on the idele group> equals the subspace topology induced by $\phi$.
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