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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.

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Can you see a reasonably easy way to show that its complement is open? – Qiaochu Yuan May 3 '11 at 3:52
@Qiaochu: I think so. Since all the shapes are circles, I think it not hard to show its complement is open using relationships between radius's and distances. But I want to know a general method when the shapes are not circles and when it is in a more general case. See the EDIT. – Roun May 3 '11 at 4:31
up vote 8 down vote accepted

It's true that $X$ is a union of closed sets, but it's also an intersection of closed sets: namely, it's the intersection of $X$ plus a closed ball of radius $\frac{1}{n}$ about the origin for all $n$. (This is the closed-set version of the open-set argument I was hinting at in the comments.)

The point here is that $X$ is locally a finite union of closed sets except at the origin, and one can "approach" the origin using the above intersection. I don't know if there's a particularly productive general statement to be made about this situation.

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