Concept

Hawaiian earring

Summary
In mathematics, the Hawaiian earring \mathbb{H} is the topological space defined by the union of circles in the Euclidean plane \R^2 with center \left(\tfrac{1}{n},0\right) and radius \tfrac{1}{n} for n = 1, 2, 3, \ldots endowed with the subspace topology: :\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 \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 \mathbb{H} is locally homeomorphic to \R at all non-origin points, \mathbb{H} is not semi-locally simply connected at (0,0). Therefore, \mathbb{H} does not have a simply connected covering space and is usually gi
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