# 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

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

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 $HR$.

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.

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