this is a question that is not addressed in my book directly but I was curious. We just proved that the union of a finite collection of closed sets is also closed, but I was curious about if the union of infinitely many closed sets can be open. This question may not be at the level of the book so perhaps that's why it wasn't addressed.
Just to make things easier, lets imagine sets that are disks in the x-y plane. I can imagine that if there are nested disks inside each other, that in this case the union would clearly be closed.
But what if you could construct an infinite set of disks that together cover the entire real plane. Then in this case, it seems that every point in their union would have an open ball centered around the point that is also contained in the real plane, so that this union of an infinite collection of disks would create an open set.
Is this a correct way of thinking? Or at least on the right track?
I get the feeling that as long as you have no largest individual set that contains all the others, then you won't get a closed set. But I have a feeling there is more subtlety to it.
Thanks everyone
Best Answer
Let $U$ be any open set whatsoever. Each point $x \in U$, taken as a one-point set $\{x\}$, is closed, and $U$ is the union of all these sets.