THE FUNDAMENTAL GROUP OF THE HAWAIIAN EARRING IS NOT FREE

Abstract

The Hawaiian earring is a topological space which is a countably infinite union of circles, that are all tangent to a single line at the same point, and whose radii tend to zero. In this note a short proof is given of a result of J.W. Morgan and I. Morrison that describes the fundamental group of this space. It is also shown that this fundamental group is not a free group, unlike the fundamental group of a wedge of an arbitrary number of circles.