## Friday, September 12, 2008

### Hawaiian Earring

This is the one thing that stuck in my mind after the two back-to-back classes yesterday. Essentially, it's the set of circles centered at (1/n,0) with radius 1/n for all n in the natural numbers (the counting numbers, n=1,2,3,...). It's actually a quite beautiful thing. Take a peak! (Thank you wikipedia!)

Now, the one thing that got me during class was that my professor contradicted himself. He first made the point that this is very similar to the wedge of circles (but that it is not homeomorphic - their topologies are very different). The difference in these two structures? Well, visually, we can see that the Hawaiian Earring are circles within circles, adjoined at one vertex. The wedge of circles (commonly referred to as the rose) are a ring of circles adjoined at one vertex. So, there's that difference. It doesn't seem that the rose contains strictly the circles of the same radius, so the size of the circles does not seem to matter.

Then it got confusing. A point was made that the Hawaiian Earring (X) is compact, and that it is not a CW-complex. But, then he went on to say that one vertex and infinitely many edges implies that the space is not compact. Yet, X has infinitely many edges and one vertex, so why is it compact?

After some research, it turns out that X is compact. And, I'm pretty certain that the rose is not compact, although I still haven't found any literature on it - I'll be hitting the library later on.

unapologetic said...

Here's a topological difference for you. On the Hawaiʻian earring, consider the points of the form \$2/n\$. This is a sequence of points, each of which is "far" from the vertex on its circle, and yet the sequence converges to the vertex.

On the infinite bouquet of circles, any sequence composed of one point "far" from the vertex on each circle does not converge.

Of course, to make real sense of this you have to give explicit meaning to "far", but that's the basic idea. Great problem, though.

plclark said...

Dear Sarah,

In order to understand the difference 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.