Hawaiian earring

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In mathematics, the Hawaiian earring ${\displaystyle \mathbb {H} }$ is the topological space defined by the union of circles in the Euclidean plane ${\displaystyle \mathbb {R} ^{2}}$ with center ${\displaystyle \left({\tfrac {1}{n}},0\right)}$ and radius ${\displaystyle {\tfrac {1}{n}}}$ for ${\displaystyle n=1,2,3,\ldots }$ endowed with the subspace topology:

$${\displaystyle \mathbb {H} =\bigcup _{n=1}^{\infty }\left\{(x,y)\in \mathbb {R} ^{2}\mid \left(x-{\frac {1}{n}}\right)^{2}+y^{2}=\left({\frac {1}{n}}\right)^{2}\right\}.}$$

The space ${\displaystyle \mathbb {H} }$ 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 ${\displaystyle \mathbb {H} }$ is locally homeomorphic to ${\displaystyle \mathbb {R} }$ at all non-origin points, ${\displaystyle \mathbb {H} }$ is not semi-locally simply connected at ${\displaystyle (0,0)}$. Therefore, ${\displaystyle \mathbb {H} }$ 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.