- #1

- 1

- 0

## Homework Statement

Consider the map phi : C -> I which maps each point of the middle third Cantor set C, considered as a subset of real numbers between 0 and 1 written in base 3 and containing only digits 0 and 2, to the set of real numbers I=[0,1] written in base 2, according to the rule: 0.a_1a_2a_3... -> 0.b_1b_2b_3... where b_i = a_i / 2

(1) Prove that phi is a continuous map of C onto I.

(2) Prove that phi is not bijective.

## Homework Equations

## The Attempt at a Solution

I decided to use the definition of continuity that all open sets in I must have open pre-images in C. I tried saying pick (1/2,1) in I (that is, all elements of the form 0.1b_2b_3...). This open intervals pre-image in C would be the intersection of (2/3,1) with the cantor set, C.

It's really sketchy in my head and I would love some help. Also proving that phi is not bijective. I feel it may have something to do with certain decimal expansions.

Thanks.