Hawaiian earring

In mathematics, the Hawaiian earring formula_1 is the topological space defined by the union of circles in the Euclidean plane formula_2 with center formula_3 and radius formula_4 for formula_5 endowed with the subspace topology:
The space formula_1 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 formula_1 is locally homeomorphic to formula_9 at all non-origin points, formula_1 is not semi-locally simply connected at formula_11. Therefore, formula_1 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 -ball around contains every circle whose radius is less than ); 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.
Fundamental group.
The Hawaiian earring is neither simply connected nor semilocally simply connected since, for all formula_13 the loop formula_14 parameterizing the th circle is not homotopic to a trivial loop. Thus, formula_1 has a nontrivial fundamental group  formula_16 sometimes referred to as the "Hawaiian earring group". The Hawaiian earring group formula_17 is uncountable, and it is not a free group. However, formula_17 is locally free in the sense that every finitely generated subgroup of formula_17 is free.
The homotopy classes of the individual loops formula_14 generate the free group formula_21 on a countably infinite number of generators, which forms a proper subgroup of formula_17. The uncountably many other elements of formula_17 arise from loops whose image is not contained in finitely many of the Hawaiian earring's circles; in fact, some of them are surjective. For example, the path that on the interval formula_24 circumnavigates the th circle. More generally, one may form infinite products of the loops formula_14 indexed over any countable linear order provided that for each formula_26, the loop formula_14 and its inverse appear within the product only finitely many times.
It is a result of John Morgan and Ian Morrison that formula_17 embeds into the inverse limit formula_29 of the free groups with generators, formula_30, where the bonding map from formula_30 to formula_32 simply kills the last generator of formula_30. However, formula_17 is a proper subgroup of the inverse limit since each loop in formula_1 may traverse each circle of formula_1 only finitely many times. An example of an element of the inverse limit that does not correspond an element of formula_17 is an infinite product of commutators formula_38, which appears formally as the sequence formula_39 in the inverse limit formula_29.
First Singular Homology.
Katsuya Eda and Kazuhiro Kawamura proved that the abelianisation of formula_41 and therefore the first singular homology group formula_42 is isomorphic to the group
formula_43.
The first summand formula_44 is the direct product of infinitely many copies of the infinite cyclic group (the Baer–Specker group). This factor represents the singular homology classes of loops that do not have winding number formula_45 around every circle of formula_1 and is precisely the first Cech Singular homology group formula_47. Additionally, formula_44 may be considered as the "infinite abelianization" of formula_17, since every element in the kernel of the natural homomorphism formula_50 is represented by an infinite product of commutators. The second summand of formula_42 consists of homology classes represented by loops whose winding number around every circle of formula_52 is zero, i.e. the kernel of the natural homomorphism formula_53. The existence of the isomorphism with formula_54 is proven abstractly using infinite abelian group theory and does not have a geometric interpretation.
Higher dimensions.
It is known that formula_52 is an aspherical space, i.e. all higher homotopy and homology groups of formula_52 are trivial.
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. The formula_57-dimensional Hawaiian earring is defined as
Hence, formula_59 is a countable union of -spheres which have one single point in common, and the topology is given by a metric in which the sphere's diameters converge]] towards zero for formula_60 Alternatively, formula_59 may be constructed as the Alexandrov compactification of a countable union of disjoint formula_62s. Recursively, one has that formula_63 consists of a convergent sequence, formula_64 is the original Hawaiian earring, and formula_65 is homeomorphic to the reduced suspension formula_66. 
For formula_67, the formula_68-dimensional Hawaiian earring is a compact, formula_69-connected and locally formula_69-connected. For formula_71, it is known that formula_72 is isomorphic to the Baer-Specker group formula_73
For formula_74 and formula_75 Barratt and Milnor showed that the singular homology groups formula_76 are nontrivialin fact, uncountable.