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Concept
Hawaiian earring
Summary
In mathematics, the Hawaiian earring is the topological space defined by the union of circles in the Euclidean plane with center and radius for endowed with the subspace topology:
The space is homeomorphic to the one-point compactification of the union of a countable family of disjoint open intervals.
The Hawaiian earring is a one-dimensional, compact, locally path-connected metrizable space. Although is locally homeomorphic to at all non-origin points, is not semi-locally simply connected at . Therefore, does not have a simply connected covering space and is usually given as the simplest example of a space with this complication.
The Hawaiian earring looks very similar to the wedge sum of countably infinitely many circles; that is, the rose with infinitely many petals, but these two spaces are not homeomorphic. The difference between their topologies is seen in the fact that, in the Hawaiian earring, every open neighborhood of the point of intersection of the circles contains all but finitely many of the circles (an ε-ball around (0, 0) contains every circle whose radius is less than ε/2); in the rose, a neighborhood of the intersection point might not fully contain any of the circles. Additionally, the rose is not compact: the complement of the distinguished point is an infinite union of open intervals; to those add a small open neighborhood of the distinguished point to get an open cover with no finite subcover.
The Hawaiian earring is neither simply connected nor semilocally simply connected since, for all the loop parameterizing the nth circle is not homotopic to a trivial loop. Thus, has a nontrivial fundamental group sometimes referred to as the Hawaiian earring group. The Hawaiian earring group is uncountable, and it is not a free group. However, is locally free in the sense that every finitely generated subgroup of is free.
The homotopy classes of the individual loops generate the free group on a countably infinite number of generators, which forms a proper subgroup of .
Official source
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