Hawaiian earring
In mathematics, the Hawaiian earring H is the topological space defined by the union of circles in the Euclidean plane R2 with center (1/n, 0) and radius 1/n for n = 1, 2, 3.... The space H is homeomorphic to the one-point compactification of the union of a countably infinite family of open intervals.
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 looks very similar to the wedge sum of countably infinitely many circles; that is, the rose with infinitely many petals, but those 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. It is also seen in the fact that the wedge sum is not compact: the complement of the distinguished point is a 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 not simply connected, since the loop parametrising any circle is not homotopic to a trivial loop. Thus, it has a nontrivial fundamental group G.
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