Depending on the sets, the union can be any of those.

Suppose A= (0, 1), an open set, and B= [-1,2], a closed set. Then their union is [-1, 2] which is closed.

Suppose A= [0, 1], a closed set, and B= (-1,2), an open set. Then their union is (-1, 2) which is open.

Suppose A= (0, 1), an open set and B= [1, 2], a closed set. Then their union is (0, 2] which is neither open nor closed.

Suppose A= , an open set, and B= , a closed set. Their union is all of R which is both open and closed.

What about them? Are you asking whether they are open or closed? In what topology? As subsets of the real numbers with the "usual" topology"?What aboutZ? What aboutQ?

In that case, Z is closed and Q is neither open nor closed. Z is closed because it has NO limit points (no sequence of integers converges) and so it is correct to say that it contains all of its limit points (the set of all of its limit points is empty and the empty set is a subset of all sets). Q is not closed because it has limit points that are not rational numbers. For example, the sequence 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, 3.141592, ..., where each number is to one more decimal place, converges to so is a limit point but is not a rational number. Q is not open because every interval contains some irrational numbers. In particular, an interval centered on any rational number, no matter how small, contains some irrational numbers so that rational number is not an "interior point".

No. While there is a theorem that says the union of a finite number of closed sets is closed, there is no theorem that says one way or the other about the union of an infinite number of closed sets. That particular example happens to be closed because its complement, U(1, 2)U(3,4)U...U(2n-1,2n)U... is a union of open sets and so open. But is a union of an infinite number of closed sets that is open- it is - while is a union of an infinite number of closed sets that is neither open nor closed- it is (0, 2]. And, of course is a union of an infinite number of closed sets which is both open and closed- it is all of R....

[0,1] U [2,3] U [4,5] U ... U [2n, 2n + 1] U ...

I assume this is an infinite collection of closed sets. Great.

If the union of a finite collection of closed sets inRis closed... does that mean this is open?

Also remember that saying a set is "not closed" does not necessarily mean it is open. It might be neither open nor closed.