  Following initial work by JaJa and Ahlswede/Cai, and inspired by a recent
renewed surge in interest in deterministic identification via noisy channels,
we consider the problem in its generality for memoryless channels with finite
output, but arbitrary input alphabets.
  Such a channel is essentially given by (the closure of) the subset of its
output distributions in the probability simplex. Our main findings are that the
maximum number of messages thus identifiable scales super-exponentially as
$2^{R\,n\log n}$ with the block length $n$, and that the optimal rate $R$ is
upper and lower bounded in terms of the covering (aka Minkowski, or Kolmogorov,
or entropy) dimension $d$ of the output set: $\frac14 d \leq R \leq d$. Leading
up to the general case, we treat the important special case of the so-called
Bernoulli channel with input alphabet $[0;1]$ and binary output, which has
$d=1$, to gain intuition. Along the way, we show a certain Hypothesis Testing
Lemma (generalising an earlier insight of Ahlswede regarding the intersection
of typical sets) that implies that for the construction of a deterministic
identification code, it is sufficient to ensure pairwise reliable
distinguishability of the output distributions.
  These results are then shown to generalise directly to classical-quantum
channels with finite-dimensional output quantum system (but arbitrary input
alphabet), and in particular to quantum channels on finite-dimensional quantum
systems under the constraint that the identification code can only use tensor
product inputs.
