§ Shrinking wedge of circles / Hawaiian earring (TODO)

I've been trying to make peace with the fact that countably infinite wedge of circles is so different from the hawaiian earring. Here are some thoughts:
  • Topology on the hawaiian earring is very different. Eg: small opens around the center of infinite wedge is contractible, while no small open around the origin of the hawaiian earring is contractible.
  • We can take infinite products of group elements in the hawaiian earring, since the radii decrease. For example, we can take a loop at each circle of radius 1/n1/n by making it traverse the circle of radius 1/n1/n in the interval t[(n1)/n,n/(n+1)]t \in [(n-1)/n, n/(n+1)], and stay at (0,0)(0, 0) at t=1t=1.