Mariano is absolutely right. [And so, for the record, is Joel.]

Coincidentally, an equivalent question came up on the blogosphere in late 2008, and I answered it. The original website is

http://mathphdthoughts.blogspot.com/2008/09/hawaiian-earring.html

Here is my post on that page:

In order to understand the difference [between the Hawaiian earring and the wedge of circles, that is -- 1/5/10] you need to look explicitly at the definition of the topology on a CW-complex with infinitely many cells. By definition, this is the direct limit over the topologies of the subcomplexes with only finitely many cells.

In other words, a subset of the CW complex is open iff its intersection with each individual cell is open. (Another way of saying this is that this is the strongest topology on the entire complex such that each of the inclusion maps from the cells is continuous. This is an instance of a "final topology." Why it is also sometimes called a "weak topology" is not so clear to me: the only reasonable explanation is that the meanings of 'weak' and 'strong' used to be the reverse of what they now are, which I believe is unfortunately the case.)

Anyway, to compare the Hawaiian earring to the infinite bouquet of circles, look at the neighborhood bases of the central point P. On the Hawaiian earring, any open set containing P must contain the entire nth circle for all sufficiently large n, and for the remaining finitely many circles must contain an open interval about P on that circle. However, on the bouquet of circles, the neighborhoods of P are exactly the subsets which contain an open interval around P on each circle. This is a much larger collection of neighborhoods, and indeed the CW-topology is strictly finer than the earring topology.

From this it is easy to see that the CW-topology is not compact, in any number of ways:

(i) Find a closed, discrete infinite subset.

(ii) Note that it is Hausdorff and apply the fact (which can be found in Rudin's *Real and Complex Analysis*) that any two compact Hausdorff topologies on the same set are incomparable.

(iii) Convince yourself that any CW-complex is compact iff it has finitely many cells.

This generated some discussion on John Armstrong's blog:

http://unapologetic.wordpress.com/2008/09/12/hawai%ca%bbian-earrings/#comments

The very last comment mentions that the earring as the one-point compactification of a countably infinite disjoint union of open intervals. This is a nice observation, the more so since **it's completely obvious**: the earring is a closed, bounded subset of the Euclidean plane, hence compact. Remove the central point from it and you do indeed get an infinite disjoint union of open intervals. (And, of course, the one-point compactification of a locally compact Hausdorff space is unique up to unique isomorphism.)

This makes clear that the homeomorphism type of the earring does not depend on the radii of the circles -- so long as they converge to 0, of course. (Note that the monotonicity is a superfluous hypothesis. If a sequence converges to $0$, you can reorder it so as to be monotonically decreasing, and the resulting subset of the plane can't tell the difference.)

for me, simple connectedness and the fundamental group is algebraic topology, not general topology. I wouldn't want to argue for this: certainly many general topology texts, e.g. Munkres's end with a chapter on this material (I would say that they end with an introduction to algebraic topology...) This is just by way of explaining my ancient comment. $\endgroup$2more comments