Answer to Question #175897 in Real Analysis for Nikhil

Question

The set Q of a rational numbers is a closed subset of R.

True or false

Answer 1

Statement is false:

The set of rational numbers Q ⊂ R is neither open nor closed.

$\bigstar$ Since any neighborhood (q−ϵ,q+ϵ)

(q−ϵ,q+ϵ) of a rational 

$\bull$ q contains irrationals, 

Q has no internal points.

$\bigstar$ This implies that Q  is not open.

$\bigstar$ Since every irrational number is the limit of a sequence of rationals, 

$\bigstar$ Q is not closed (for a set to be closed it should contain all of its limit points).

$\bigstar$ Since every one-point-set {x}⊂R

{x}⊂R is closed,

$\bull$ and since 

Q is countable, we have that

$\boxed{Q=U_{p∈Q }{P} }$

is a countable union of closed sets.