The Hawaiian Earring $X$ is the union of the circles $[x-(1/n)]^2+y^2=(1/n)^2,n=1,2,3...$ with the topology from the plane.
I want to show that $X$ is closed.
I note that $X$ is a countable union of closed sets, which is not necessarily closed. However, I've saw a theorem like this:
The union of a locally finite collection of closed sets is closed.
But again the Hawaiian Earring is not a locally finite collection of closed sets.
I know there may be some problem-specific proofs. But I want to know whether there is a general theorem like the one above that shows $X$ is closed, because I feel that there are something common in this problem but I can't figure it out.
Can you please help? Thank you!
EDIT: I want to know if there is a theorem of the following kind.
When a collection of closed sets (may be infinitely) satisfied condition XXX, then the union of them are still closed.