  Dag Normann and the author have recently initiated the study of the logical
and computational properties of the uncountability of $\mathbb{R}$ formalised
as the statement $\textsf{NIN}$ (resp. $\textsf{NBI}$ that there is no
injection (resp. bijection) from $[0,1]$ to $\mathbb{N}$. On one hand, these
principles are hard to prove relative to the usual scale based on comprehension
and discontinuous functionals. On the other hand, these principles are among
the weakest principles on a new complimentary scale based on (classically
valid) continuity axioms from Brouwer's intuitionistic mathematics. We continue
the study of $\textsf{NIN}$ and $\textsf{NBI}$ relative to the latter scale,
connecting these principles with theorems about Baire classes, metric spaces,
and unordered sums. The importance of the first two topics requires no
explanation, while the final topic's main theorem, i.e. that when they exist,
unordered sums are (countable) series, has the rather unique property of
implying $\textsf{NIN}$ formulated with the Cauchy criterion, and (only)
$\textsf{NBI}$ when formulated with limits. This study is undertaken within
Ulrich Kohlenbach's framework of higher-order Reverse Mathematics.
