9.2 Cantor set

Let in the following \(C⊂ℝ\) be Cantor’s ternary set. This set is described in many texts, as for example Sect. 2.44 in [ 22 ] ; and also in Wikipedia [ 51 ] ).

E274

[09S](Replaces 0W4) Show that \(C\) is closed, and composed only of accumulation points. Hence \(C\) is a perfect set.

E274

[09T] Let \(I=\{ 0,2\} \) and \(X=I^ℕ\), consider the map \(F:X→ C\) given by

\[ F(x)=∑_{n=0}^∞ 3^{-n-1} x_ n~ . \]

Show that it is a bijection.

Let’s now equip \(X\) with the topology defined in 266.  1 . Show that \(F\) is a homeomorphism.

Hidden solution: [UNACCESSIBLE UUID ’09V’]

See also 2, 4, 3.

[UNACCESSIBLE UUID ’09W’]

  1. Note that the order topology on \(I=\{ 0,2\} \) is also the discrete topology.