Topology on fractional ideals of a number field

Question

Is there a natural (in some suitable sense) non discrete topology on the group of all non-zero fractional ideals of a number field?

Answer 1

The most natural topology is surely the subspace topology from the idele group of the global field, this is part of how we manage the surjection of $k_{\Bbb A}^\times\to Cl_k$ of the ideles onto the class group. The embedding is as usual given by letting $S$ be the set of (finite) primes of $k$ and mapping an ideal into $k_{\Bbb A}^\times$ via

$$\mathfrak{a}\mapsto (1, 1, \ldots, 1)_\infty\times (\pi_p^{v_p(\mathfrak{a})})_{p\in S}$$

where $v_p$ is the valuation for the prime, $p$, and the $1$s are from the infinite places and the $\pi_p$ are local uniformizers at $p$, i.e. for the completion $k_p$.