EDB — 2F3

[2F3]Topics:perfect set.Prerequisites:[0QP]↺↻,[2F2]↺↻,[2FD]↺↻.Difficulty:**.

Suppose \((X,d)\) is a complete metric space. A closed set \(E⊆ X\) without isolated points, i.e. consisting only of accumulation points, is called a perfect set.

Let \(C\) be the Cantor set. Assume that \(E\) is perfect and non-empty. Show that there exists a continuous function \(𝜑:C→ E\) that is an homeomorphism with its image. This implies that \(|E|≥ |ℝ|\).

So, in a sense, any non-empty perfect set contains a copy of the Cantor set.

This can be proven without relying on continuum hypothesis [2F2]↺↻. Cf. [0W3]↺↻.

Due to [0J8]↺↻, it is enough to show that there exists a \(𝜑:C→ E\) continuous and injective.