  Motivated by results of J. R. Kline and R. L. Moore (1919) that a compact
subset of the plane, homeomorphic to a subset of the reals, lies on the arc, we
give a purely topological characterisation of compact sets of the reals. This
allows us to reduce investigations of Cantorvals to properties of countable
linear orders and to show, applying the Mazurkiewicz--Sierpi\'nski Theorem
(1920), that there exist continuum many non-homeomorphic L-Cantorvals.
