A- Find an example of a disconnected set whose closure is connected.
b- If A is connected ,is A closure necessarily connected? If A is perfect is A closure necessarily perfect?© BrainMass Inc. brainmass.com December 24, 2021, 5:05 pm ad1c9bdddf
a. Q is the set of rational numbers. It is obviously a disconnected set because we can find an irrational number a, such that Q is contained in the union of the two disjoint open sets (-inf,a) and (a,inf). Because Q is dense in R, so the closure of Q is R, which is connected.
(1) Yes. If A is connected, the A ...
Connected and Disconnected Sets and Closure are investigated. The solution is detailed.