On compact subsets of the reals

Abstract

The result of Moore and Kline (1919) [10] says that a compact subset of the plane homeomorphic to a subset of the reals lies on the arc. Motivated by this result, we give a purely topological characterization 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–Sierpiński Theorem (Mazurkiewicz and Sierpiński (1920) [9]), that there exist continuum many non-homeomorphic L-Cantorvals.