3.4 *The Cantor Set 175
(Project) Fat Cantor-like Sets: For "fat" Cantor-like sets C(a) de-
fined in Definition 3.4.23, which of Theorems 3.4.2 and 3.4.3, Lemma
3.4.4, and Theorems 3.4. 12 , 3.4.15, and 3.4.17 remain true?
Generalized Cantor Sets: Notice that the Cantor set and "Cantor-like"
sets are nonempty, bounded, perfect, and nowhere dense. A set with these
properties is called a generalized Cantor set. Prove that the union of
a finite co llection of generalized Cantor sets is a generalized Cantor set.