Show that the set of rational numbers in 0 1 is not connecte

Show that the set of rational numbers in [0, 1] is not connected.

Solution

two sets which are open with respect to subspace topology and their union is the set of rational numbers in [0,1]2

choose any irrational in [0,1]

say r = sqrt2/2

A= Q intersection [0,r)

B = Q intersection (r,0]

A and B are open

and non empty

and their union is the set Q intersection [0,1]

So Q intersection [0,1] is not connected