  We extend a recently proposed 1-nearest-neighbor based multiclass learning
algorithm and prove that our modification is universally strongly
Bayes-consistent in all metric spaces admitting any such learner, making it an
"optimistically universal" Bayes-consistent learner. This is the first learning
algorithm known to enjoy this property; by comparison, the $k$-NN classifier
and its variants are not generally universally Bayes-consistent, except under
additional structural assumptions, such as an inner product, a norm, finite
dimension, or a Besicovitch-type property. The metric spaces in which universal
Bayes consistency is possible are the "essentially separable" ones -- a notion
that we define, which is more general than standard separability. The existence
of metric spaces that are not essentially separable is widely believed to be
independent of the ZFC axioms of set theory. We prove that essential
separability exactly characterizes the existence of a universal
Bayes-consistent learner for the given metric space. In particular, this yields
the first impossibility result for universal Bayes consistency. Taken together,
our results completely characterize strong and weak universal Bayes consistency
in metric spaces.
