Hawaiian earring

is the topological space defined by the union of circles in the Euclidean plane

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.

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.

parameterizing the nth circle is not homotopic to a trivial loop.

sometimes referred to as the Hawaiian earring group.

is locally free in the sense that every finitely generated subgroup of

The homotopy classes of the individual loops

on a countably infinite number of generators, which forms a proper subgroup of

arise from loops whose image is not contained in finitely many of the Hawaiian earring's circles; in fact, some of them are surjective.

More generally, one may form infinite products of the loops

indexed over any countable linear order provided that for each

and its inverse appear within the product only finitely many times.

It is a result of John Morgan and Ian Morrison that

is a proper subgroup of the inverse limit since each loop in

Katsuya Eda and Kazuhiro Kawamura proved that the abelianisation of

This factor represents the singular homology classes of loops that do not have winding number

and is precisely the first Cech Singular homology group

, since every element in the kernel of the natural homomorphism

is represented by an infinite product of commutators.

consists of homology classes represented by loops whose winding number around every circle of

is proven abstractly using infinite abelian group theory and does not have a geometric interpretation.

is an aspherical space, i.e. all higher homotopy and homology groups of

The Hawaiian earring can be generalized to higher dimensions.

Such a generalization was used by Michael Barratt and John Milnor to provide examples of compact, finite-dimensional spaces with nontrivial singular homology groups in dimensions larger than that of the space.

is a countable union of k-spheres which have one single point in common, and the topology is given by a metric in which the sphere's diameters converge to zero as

may be constructed as the Alexandrov compactification of a countable union of disjoint

Barratt and Milnor showed that the singular homology group