Question #115205
Show that the set of rational numbers with the subspace topology of R is disconnected
1
Expert's answer
2020-05-12T18:44:42-0400

Let a\in R be an irrational number.

Then a\notin Q (rationals)

Let us suppose the sets :

P = (- \infty , a ) \cap Q

T = ( a, \infty ) \cap Q


Let x P\in P

Let B(\in ,x) be an open ball in of x in Q

Then, for all x in P, there exists \in from R such that B(\in ,x) lies in P if =ax\in = a-x


Similarly, for all x in T , there exists \in from R such that B (\in , x ) lies in T if =xa\in = x-a


So , there open neighborhoods of P and T in Q, hence P and T are open sets in Q.

Now,

P \cup T = Q , P\cap T = \varnothing where P and T are non-empty open sets.

So, P and T are a separation in Q.

Hence, the result.



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