The first statement is true but how you prove it depends entirely on how you DEFINE "closed

set" and there are several equivalent ways. For example, it is possible to define "closed set" as

being a member of a collection of all subset of the universal set U such that

1) U is in the collection

2) The empty set is in the collection

3) The intersection of any family of sets in the collection is in the collection

4) The union of anyfinitefamily of sets in the collection is in the collection.

In this case, you theorem is just (3) in the definition of "closed"! (This particular definition is not

often used.)

What definition of "closed set" are you using?

As for the second, yes it is true and you only need to give a single counter-example. Consider

for all integers n> 1. That is

,

,

You will need to show that each of those sets is closed, that the union of all such sets is (0, 1),

and that (0, 1) is not closed.

(Note that proving it is open is NOT proving it is closed.)

Here [a,b] is the closed interval and (a, b) is the open interval in the real numbers.