Let a R be an irrational number.
Then a Q (rationals)
Let us suppose the sets :
P = (- , a ) Q
T = ( a, ) Q
Let x
Let B( ,x) be an open ball in of x in Q
Then, for all x in P, there exists from R such that B( ,x) lies in P if
Similarly, for all x in T , there exists from R such that B ( , x ) lies in T if
So , there open neighborhoods of P and T in Q, hence P and T are open sets in Q.
Now,
P T = Q , P T = where P and T are non-empty open sets.
So, P and T are a separation in Q.
Hence, the result.
Comments