Question

give an example to show that the union of an infinite collection of closed sets in a metric space is not necessarily closed​

Answer 1

Answer:

An example for a countably infinite union of closed sets that is not closed is Q=∪r{r} where r is an enumeration of Q. ... Each of these sets is clearly closed. However, their union over all N is (0,2] which is neither open nor closed. P.S: Note that the union cannot contain 0 because there is no n∈N such that 1n=0.