Hawaiian Earring vs Wedge Sum - A question of definition Several earlier articles have explained the topological difference between a countably infinite bouquet of circles vs. a countably infinite Hawaiian Earring of circles.  But my question is purely one of definitions.
What has not been adequately answered in earlier articles (although frequently asked) is why the Hawaiian Earring does not fit the definition of a wedge sum.
For example, on p49 Hatcher states "one might confuse X [the Hawaiian Earring] with the wedge sum of an infinite number of circles."  Yet, using Hatcher's definition of wedge sums (p10), the Hawaiian Earring appears to be a wedge sum of an infinite number of circles.
So is the Hawaiian Earring a type of wedge sum, or not?
 A: A way to see it is that the wedge sum of infinitely many circles is not compact while the Hawaiian earring is compact (It is bounded and it contains its limit points, so closed, and therefore you can use Heine-Borel) (you can even see it as one point compactification of infinitely many intervals.)
Also Hatcher gives a reason using the fundamental group. While in wedge sums you can use Van Kampen Theorem to obtain that the fundamental group is free. In the case of the Hawaiian earring the fundamental group is bigger as he explains.
In what matters to the topology, the wedge sum can be constructed as a quotient space while the Hawaiian earring has the subspace topology from $\mathbb{R}^2$. And they are not homeomorphic. In fact, the topology of the wedge sum of infinitely many circles is finer.
I hope this helps. If you suggest me another approach I could try to give other arguments.
See this:
page of book on google books related
And this article could very helpful:
article
EDIT:
In what matters to the definition, the wedge sum is constructed as a quotient space (you don't need the "ambient" space $\mathbb{R}^2$ and it is a topological space "by its own right") while the Hawaiian earring has the subspace topology from $\mathbb{R}^2$, it is not given a quotient topology. And they are not homeomorphic. In fact, the topology of the wedge sum of infinitely many circles is finer. I mean, the set is the same but the topologies are different.
A: In the Hawaiian earring, every open neighborhood of the origin contains all but a finite number of circles. No point in the wedge of circles has this property. Therefore they cannot be homeomorphic.
