Suppose we take all the rationals and take any neighbourhood around each of them. Will they cover wh

Brennen Fisher

Brennen Fisher

Answered question

2022-05-18

Suppose we take all the rationals and take any neighbourhood around each of them. Will they cover whole R . Rationals are dense. So, for each irrational we can find a rational which is arbitrary close to it. Is it correct ?

Answer & Explanation

Tyree Duke

Tyree Duke

Beginner2022-05-19Added 10 answers

No to the first statement, yes to the second.
there are only countably many rationals, so we can enumerate them in some order q 1 , q 2 , etc. Now let us look at the set:
U := i N B ( q i , 2 i 1 )Where B ( q , r ) is the open ball with center q and radius r. This set has measure 0.5 (which you can see by summing up all the radii), is open and contains all rational numbers.
So despite the fact that the rationals are dense in R , there are very small open neighborhoods of all the rationals.

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