  A 1910 theorem of Brouwer characterizes the Cantor set as the unique totally
disconnected, compact metric space without isolated points. A 1920 theorem of
Sierpinski characterizes the rationals as the unique countable metric space
without isolated points. The purpose of this exposition is to give an
accessible overview of this celebrated pair of uniqueness results. It is
illuminating to treat the problems simultaneously because of commonalities in
their proofs. Some of the more counterintuitive implications of these results
are explored through examples. Additionally, near-examples are provided which
thwart various attempts to relax hypotheses.
