no continuous function switches the rational and the irrational numbers
Let denote the irrationals. There is no continuous function such that and .
Suppose there is such a function .
But , so is first category too. Therefore is first category, as . Consequently, we have .
But functions preserve unions in both ways, so
Now, is continuous, and as is closed for every , so is . This means that . If , we have that there is an open interval , and this implies that there is an irrational number and a rational number such that both lie in , which is not possible because this would imply that , and then would map an irrational and a rational number to the same element, but by hypothesis and .
Then, it must be for every , and this implies that is first category (by (1)). This is absurd, by the Baire Category Theorem.
|Title||no continuous function switches the rational and the irrational numbers|
|Date of creation||2013-03-22 14:59:15|
|Last modified on||2013-03-22 14:59:15|
|Last modified by||yark (2760)|