Yes that is true. Both the question and the answers given are well within general topology. But the only issue is that general topologists don't worry about this space.

What counterexample in general topology did you use the Hawaiian ear ring for? I would like to know.

If this is a classic one, where else did you see it?

I also ask about Hanche-Olsen's objection.

IS the monodromy theorem relevant for you?

What is the difference between BG and the Eilenberg-Maclane space K(G,1) mentioned in Allen Hatcher's book?

@Ilya. I took a look at page 16. That is Chapter 2 on Faltings theorem, and I had imagined that it was a separate article. Read on, you will find that Faltings settled that too. The role of Shafarevich is that he proved it for elliptic curves -- the proof is rather easy, it can be found in Silverman's book on elliptic curves. The higher dimensional case was then called "Shafarevich conjecture".

This is a comment about the tag. Nobody studies Hawaiian earring in general topology. The first encounter is when you study algebraic topology -- more precisely, the fundamental group.

Faltings proved that also, along with the Mordell conjecture. See Darmon's article on Faltings' theorem.

Why isn't the following used: $g(z)$ has the unit circle as natural boundary, because its derivative $f(z)$ blows up at all roots of unity and has the unit circle as the natural boundary.

@Mariano. Supposing a_n goes to infinity. Then we should get the wedge. If a_n goes to zero, then it should be the Hawaiian. No?

Oops, I screwed by deleting my comment and re-posting in a different form.. The one of mine above, should have been the first. Sorry.

That is the wedge of infinitely many circles. Isn't that non-homeomorphic to the Hawaiian?

You could have replaced $n!$ by $n^n$ for instance, and then called it your own!