example of a space that is not semilocally simply connected

An example of a space that is not semilocally simply connected isthe 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 connectedneighborhood. 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 topologyinduced by its inclusion in the plane. In contrast, the same set endowed withthe CW topology is just a bouquet of countably many circles and (as any CWcomplex) it is semilocaly simply connected.