Is irrational number set on <mrow class="MJX-TeXAtom-ORD">

Roland Waters

Roland Waters

Answered question

2022-06-06

Is irrational number set on R disconnected?

Answer & Explanation

victorollaa5

victorollaa5

Beginner2022-06-07Added 16 answers

Consider the sets ( , 0 ) Q and ( 0 , ) Q . Each of these sets is a non-empty open subset of R Q and they are disjoint. Therefore, R Q is disconnected.
George Bray

George Bray

Beginner2022-06-08Added 12 answers

Not only are the Rationals disconnected but they are totally disconnected. Every single Rational qi gives rise to a disconnection ( , q i ) , ( q i , + ) so that connected components are singletons. Any neighborhood of the Irrationals can be disconnected in this way.

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