Let I = (R\Q)∩[0,1] with its usual metric. Prove that there is a continuous function g mapping I onto Q∩[0,1]

Answer 1

Let N = {1,2,3,...} be the set of natural numbers with the topology induced by R (so N is discrete). Take f: I --> N defined by:

f(x) = n

whenever x is in I∩(1-(2^{n-1}), 1-(2^n)). Thus f is clearly a surjection. Moreover f is continuous since, given any subset U of N, f^{-1}(U) is a union of intervals of I. Now take any surjection h: N --> Q∩[0,1] (h exists since N and Q∩[0,1] have the same cardinality), which is surely continuous since N is discrete, and define g to be the composition:

g := h o f

Then g is continuous as a composition of continuous maps, and it is surjective as a composition of surjections.

Answer 2

I woodn't think i'm an original,the world's been turning for a long time before I was born.Might have seen the original scheme of things. Wood confess to being individual in my thinking tho.Not sure how I wood prove it,maybe dress up in a tigger costume in the 100 acre wood. : )

Answer 3

Have you looked at using the Tietze Extension Theorem?