Continuous onto function from $(\Bbb R\setminus \Bbb Q)\cap [0,1]\to \Bbb Q\cap [0,1]$

Asked Dec 13 '16 at 06:21

Active Dec 13 '16 at 07:16

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Does there exist a continuous onto function from :

$(\Bbb Q)\cap [0,1]\to \Bbb R\setminus \Bbb Q\cap [0,1]$?
$(\Bbb R\setminus \Bbb Q)\cap [0,1]\to \Bbb Q\cap [0,1]$?

(1) is false because $(\Bbb Q)\cap [0,1]$ is countable whereas $ \Bbb R\setminus \Bbb Q\cap [0,1]$ is not and image of a countable set is countable.

(2) .I don't know about this.Please give some hints on this question.I am clueless here.

EDIT:In the question given as duplicate,I am unable to understand how to map each irrational number to a rational number.Please give some details.

edited Dec 13 '16 at 06:42

asked Dec 13 '16 at 06:21

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Possible duplicate of Let $f:(\mathbb{R}\setminus\mathbb{Q})\cap [0,1]\to \mathbb{Q}\cap [0,1]$. Prove there exists a continuous$f$. – Mark Schultz-Wu Dec 13 '16 at 06:27
How did you get the duplicate,I couldn't@Mark – Learnmore Dec 13 '16 at 06:33
Do you mean how did I find it? It was the 7th link down under "related" for me. Or do you want an explaination of how the duplicate solves the question? – Mark Schultz-Wu Dec 13 '16 at 06:35
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No I meant in the answer provided ,how to properly define that function – Learnmore Dec 13 '16 at 06:39
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Also a possible duplicate of this. That answer has a much better construction than the first I linked. – Mark Schultz-Wu Dec 13 '16 at 07:32
All good things come in three. – user60589 Dec 13 '16 at 10:27

Continuous onto function from $(\Bbb R\setminus \Bbb Q)\cap [0,1]\to \Bbb Q\cap [0,1]$

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