Homework SourseShow that the real numbers under the discrete topology conta
Show that the real numbers under the discrete topology contains uncountable subsets in which no point is a limit point.
Show that the real numbers under the discrete topology contains uncountable subsets in which no point is a limit point.
Solution
The proof is as follows. Suppose that A is a discrete space with a countable basis (Bj:jN). We want to prove A is countable. Discreteness of A means that each aA is in some basic open set that contains aa and no other point in A. Thus we can make a map ff which sends each aA to the least jN such that Bj={a}. This is an injection from A to N, so A is countable.
