Hawaiian earring
In mathematics, the Hawaiian earring is the topological space that arises by considering the one-point compactification of a countably infinite family of open intervals. Alternatively, consider a null sequence rn and the union of circles with center (0, rn) and radius rn in the Euclidean plane. The Hawaiian earring can be given a complete metric and it is compact. It is path connected but not semilocally simply connected.
The Hawaiian earring is not simply connected, since the loop parametrising the outermost circle is not homotopic to a trivial loop. Thus, it has a nontrivial fundamental group G. The Hawaiian earring looks very similar to the wedge (topology) of countably infinitely many circles, which has the free group of countably infinitely many generators as its fundamental group. In fact, we can embed this free group into G, but G is larger. It contains additional elements which arise from loops whose image is not contained in finitely many of the Hawaiian's earrings circles; in fact, some of them are surjective. It has been shown that G embeds into the inverse limit of the free groups with n generators, Fn, where the map from Fn to Fn+1 is just the one sending the generators of Fn to the first n generators of Fn+1. However G is not the complete inverse limit but rather the subgroup in which each generator appears only finitely many times. An example of an element of the inverse limit which is not an element of G is an infinite commutator. G is uncountable, and it is not a free group. While its Abelianisation has no known simple description, it has a normal subgroup N such that , the direct product of infinitely many copies of the infinite cyclic group.
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