The (countably infinite) wedge sum of circles is quotient of disjoint countable union of circles $\amalg S_i$, with points $x_i\in S_i$ identified to a single point, while the Hawaiian earring H is the topological space defined by the union of circles in the Euclidean plane $\mathbb{R}^2$ with center $(1/n, 0)$ and radius $1/n$ for $n = 1, 2, 3, ...$.
In the definition of countably infinite wedge sum of circles, it is not specified the size of circles, the points which to be identified to a single point etc. So we can take disjoint union of circles of radius $1/n$ and identify a point from each to a common single point to get Hawaiian earring.
I couldn't understood the difference between these two topological spaces. Can one explain more precisely the difference between these two spaces?
Answer
You have two very nice answers discussing the difference between the topologies on these spaces. However, I thought I'd mention one slightly higher-level difference between them. The fundamental group of the wedge of infinitely many circles is the free group on countable many generators, one for each circle. This is a rather uncomplicated countable group. The fundamental group of the Hawaiian earring, however, is truly bizarre. In fact, it is uncountable and has many rather complicated relations in it.
When I first learned about this, I was shocked that closed subsets of the plane could have uncountable fundamental groups.
A nice paper that discusses this (and contains a good bibliography of earlier work) is "The combinatorial structure of the Hawaiian earring group" by Cannon and Conner, which appeared in Topology and its Applications, Volume 106, Issue 3, 6 October 2000, Pages 225-271.
No comments:
Post a Comment