Well, if you set Sn=[-n,n], then take the union of each Sn for n=1,2,3,4...
Each [-n,n] is clearly compact. But the union (which is R itself), is not.
Find an infinite collection {Sn : n in N} of compact sets in R such that infinite union of Sn is not compact.
Is it correct to say that the union of infinite sets is a countable infinity?
So that if the union of infinite sets is going to be bounded by N, but you could find a set N + 1 that is not in the set?
Be careful here. If you have a set then clearly is countable for the "worst case scenario" is that all the sets are distinct in which case the mappking given by is clearly a bijection. Saying that given a countable class of sets the union is automatically countable is just plain wrong. What about ?