Descriptions of Cantor Sets: A Set‑Theoretic Survey and Open Problems

This paper synthesizes the principal descriptive set-theoretic perspectives on deterministic Cantor sets on the real line and charts directions for future study. After recounting their historical genesis and compiling an up-to-date taxonomy, we review the Borel hierarchy and four hierarchically ordered representations, general, nested, iterated-function-system (IFS), and q-ary expansion, presented from the most general to the most specific set-theoretic description of deterministic Cantor sets. We then present explicit closed-form descriptions for two thin families of measure-zero Cantor sets and, for the augmented "thick" family of positive Lebesgue measure, a new closed recursive formula for the endpoint maps governing every retained interval at every finite stage; we further show that the classical middle-third set lies in the intersection of all three families, providing a unifying example across the four representations. The survey closes by isolating several open problems in four directions, aiming to provide mathematicians with a coherent platform for further descriptive set-theoretic investigations into Cantor-type sets on the real line.