Answer to Question #114846 in Real Analysis for Sheela John

Question

Show that the set of rational numbers with the subspace topology of R is disconnected

Answer 1

Let a∈ R be an irrational number.

Then a$\notin$ Q (rationals)

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Let us suppose the sets :

P = (- ∞ , a ) ∩ Q

T = ( a, ∞ ) ∩ Q

Let x ∈P

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 

∈=a−x

Similarly, for all x in T , there exists ∈ from R such that B (∈ , x ) lies in T if 

∈=x−a

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.