called the big fundamental
group, which satisfies
.Abstract: The Hawaiian earring H is the union of the countably many planar circles Cn, with n = 1, 2, ...,
where Cn has radius 1/n and is tangent to the x-axis at the origin. The earring H is famous since
it has no simply connected covering space and since its fundamental group BF is neither countable nor
free. We shall give natural descriptions of H and BF that reveal them as the countable cases of
constructions H(C) and BF(C) valid for every cardinal C, H(C) being the generalized Hawaiian
earring and BF(C) the big free group. These generalized objects are connected by a new topological
invariant called the big fundamental
group, which satisfies
.