Take the set of points 1/n for all integers n. This is an isolated set of points- that is, for any of them there is an open ball about the point not containing any other point. However, this set has a limit point which is not contained in the set (namely 0), hence it is not closed.
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A singleton point is a closed set. The natural numbers can be written as a countable union of points. Thus, they form a Borel set.
A graph composed of isolated points.
The set of even numbers is closed under addition, the set of odd numbers is not.
That is correct, the set is not closed.
The solution set is all points on the circle.