example of a space that is not semilocally simply connected

An example of a space that is not semilocally simply connected is the following: Let

$$HR=\bigcup _{n\in \mathbb{N}}\left\{(x,y)\in {ℝ}^2\right|{\left(x-\frac{1}{2^n}\right)}^2+y^2={\left(\frac{1}{2^n}\right)}^2\}$$

endowed with the subspace topology. Then $(0,0)$ has no simply connected neighborhood. Indeed every neighborhood of $(0,0)$ contains (ever diminshing) homotopically non-trivial loops. Furthermore these loops are homotopically non-trivial even when considered as loops in $HR$.

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The Hawaiian rings

It is essential in this example that $HR$ is endowed with the topology induced by its inclusion in the plane. In contrast, the same set endowed with the CW topology is just a bouquet of countably many circles and (as any CW complex) it is semilocaly simply connected.

Title	example of a space that is not semilocally simply connected
Canonical name	ExampleOfASpaceThatIsNotSemilocallySimplyConnected
Date of creation	2013-03-22 13:25:10
Last modified on	2013-03-22 13:25:10
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	16
Author	mathcam (2727)
Entry type	Example
Classification	msc 57M10
Classification	msc 54D05
Defines	Hawaiian rings
Defines	Hawaiian earrings