For questions concerning the Cantor set, which consists of those real numbers in $ [0,1]$ that remain after repeatedly removing the open middle third of every interval; it contains those numbers which may be written in ternary without using 1. Also, for questions about other topological spaces that are homeomorphic to the Cantor set.
The Cantor set, also known as Cantor's "middle thirds" set, consists of those real numbers that remain when one starts with the range [0,1] and repeatedly removes the open middle third of every remaining interval; it contains those numbers which may be expressed in ternary without using 1, ee.g. $0$, $1/3 = 0.1 = 0.0\dot{2}$, $2/3 = 0.2$, $1 = 0.\dot{2}$.
The Cantor set is a closed subset of $[0,1]$ and is a compact topological space. It has Lebesgue measure $0$, is nowhere dense, yet it is uncountable. It displays fractal self-similarity.
The construction can be generalized to "fat" Cantor sets whose Lebesgue measure is positive (and can be any possible value in the interval $(0,1)$). Fat Cantor sets are still nowhere dense, witnessing that a nowhere dense set can have positive Lebesgue measure.
Topological spaces homeomorphic to Cantor's middle thirds set are known simply as "Cantor sets" or "Cantor spaces". An important example is the product $\lbrace 0,1\rbrace^\mathbb{Z}$ of countably many copies of the discrete space $\lbrace 0,1\rbrace$. Cantor spaces can be characterized as the nonempty, compact, totally disconnected, metrizable spaces having no isolated points.
