Hawaiian Earring

Hey all,

I'm wondering if anyone can help me with the Hawaiian earring.
Basically, I don't understand why it is compact.

X is a union of (x-1/n)^2 + y^2 = (1/n)^2 with the subspace topology:

I thought that the subspace topology meant that you can view each
circle as an open set, as you can intersect each circle with an open
set in the plane. So, the Hawaiian earring is a union of these open
sets, so therefore the collection of circles forms a cover for X. What
I thought is that if you remove one of these open sets, you won't have
a subcover, and therefore it is not compact.

Clearly, I am misunderstanding something rather fundamental...

If anyone can help, that would be great!

Natalie

.

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